> ## Documentation Index > Fetch the complete documentation index at: https://docs.iotools.cloud/llms.txt > Use this file to discover all available pages before exploring further. # Infinite Series Sum Calculator API > Compute the closed-form sum of a geometric, p-series, telescoping, alternating harmonic, Leibniz, Basel, exponential (e^x), or alternating p-series — with the exact convergence reason, formula, and a partial-sums table showing how quickly (or slowly) it converges. [Try Infinite Series Sum Calculator in your browser →](https://iotools.cloud/tool/infinite-series-sum-calculator/) ## OpenAPI ````yaml https://api.iotools.cloud/v1/openapi post /v1/tool/infinite-series-sum-calculator openapi: 3.1.0 info: title: iotools.cloud API version: 1.0.0 description: >- Run any iotools.cloud tool over HTTP. Authenticate with `Authorization: Bearer iot_live_…`. **Only `POST /v1/tool/{slug}` costs credits.** Every GET here — the catalog, a tool's schema, your balance — is free. A tool call is charged its own weight or your plan's per-call minimum, whichever is larger; `x-iotools-credit-cost` on each operation is quoted at the free-tier minimum, and `GET /v1/tools/list` returns the exact figure for your key. `GET /v1/me/credits` reports what you have left, and `GET /v1/me/usage` reports what it went on. Failures are RFC 9457 problem documents — branch on `code`. servers: - url: https://api.iotools.cloud description: Production security: - bearerAuth: [] tags: - name: Catalog description: Find a tool and read its contract. Free. - name: Converters description: >- Convert between formats, encodings, and units — Base64, CSV and JSON, timestamps, and more. Fast, free, and processed right in your browser. - name: Formatters description: >- Format, beautify, minify, and validate code and data — JSON, HTML, CSS, SQL, and regex. Clean up messy input in one click, with nothing to upload. - name: Generators description: >- Generate exactly what you need and on demand — passwords, UUIDs, QR codes, hashes, random numbers, and more. Secure, instant, and free. - name: Calculators description: >- Crunch the numbers fast — from everyday math to specialized conversions and unit work. Free online calculators that run entirely in your browser. - name: Editors description: >- Edit and transform text, code, and images with quick, focused editors that run entirely in your browser — nothing to install, and no sign-up needed. - name: Utilities description: >- Everyday developer and web utilities — DNS and IP lookups, redirect and certificate checkers, and other quick diagnostics. Free and privacy-friendly. - name: Account description: Your key's allowance, limits and usage history. paths: /v1/tool/infinite-series-sum-calculator: post: tags: - Calculators summary: Infinite Series Sum Calculator description: >- Compute the closed-form sum of a geometric, p-series, telescoping, alternating harmonic, Leibniz, Basel, exponential (e^x), or alternating p-series — with the exact convergence reason, formula, and a partial-sums table showing how quickly (or slowly) it converges. [Try Infinite Series Sum Calculator in your browser →](https://iotools.cloud/tool/infinite-series-sum-calculator/) operationId: run_infinite_series_sum_calculator requestBody: required: false content: application/json: schema: type: object properties: series: enum: - geometric - p_series - telescoping - alt_harmonic - leibniz - basel - exp_x - alt_p description: Series family a: type: number description: First-term coefficient a r: type: number description: Common ratio r p: type: number description: Exponent p k: type: number description: Gap k minimum: 1 x: type: number description: Argument x n0: type: number description: Starting index n₀ minimum: 1 maximum: 50 numTerms: type: number description: Partial sums to show minimum: 5 maximum: 30 required: [] examples: geometric_series_a_1_r_0_5_converges: summary: Geometric series, a = 1, r = 0.5 (converges) value: series: geometric a: '1' r: '0.5' n0: '1' numTerms: '5' p_series_p_2_the_basel_problem_via_2: summary: p-series, p = 2 — the Basel problem via ζ(2) value: series: p_series p: '2' numTerms: '5' leibniz_series_for_very_slow_convergence: summary: Leibniz series for π (very slow convergence) value: series: leibniz numTerms: '5' telescoping_series_k_2_n_1: summary: Telescoping series, k = 2, n₀ = 1 value: series: telescoping k: '2' n0: '1' numTerms: '5' responses: '200': description: Tool output content: application/json: examples: geometric_series_a_1_r_0_5_converges: summary: Geometric series, a = 1, r = 0.5 (converges) value: tool: infinite-series-sum-calculator tool_version: 1.0.2 outputs: converges: Converges exactSum: 1·0.5^1 / (1 - 0.5) numericSum: '1' formula: S = a·rⁿ⁰ / (1 - r) reason: '|r| = 0.5 < 1, so the geometric series converges.' partialSums: - 'n': '1' an: '0.5' sn-partial-sum: '0.5' - 'n': '2' an: '0.25' sn-partial-sum: '0.75' - 'n': '3' an: '0.125' sn-partial-sum: '0.875' - 'n': '4' an: '0.0625' sn-partial-sum: '0.9375' - 'n': '5' an: '0.03125' sn-partial-sum: '0.96875' steps: >- Series: Geometric series (Σ a·rⁿ) Convergence: Converges Reason: |r| = 0.5 < 1, so the geometric series converges. Formula: S = a·rⁿ⁰ / (1 - r) Exact sum: 1·0.5^1 / (1 - 0.5) Numeric sum: 1 Partial sums: S(1) = 0.5 (aₙ = 0.5) S(2) = 0.75 (aₙ = 0.25) S(3) = 0.875 (aₙ = 0.125) S(4) = 0.9375 (aₙ = 0.0625) S(5) = 0.96875 (aₙ = 0.03125) credits_used: 5 credits_remaining: null p_series_p_2_the_basel_problem_via_2: summary: p-series, p = 2 — the Basel problem via ζ(2) value: tool: infinite-series-sum-calculator tool_version: 1.0.2 outputs: converges: Converges exactSum: π²/6 numericSum: '1.64493' formula: S = ζ(p) (Riemann zeta function) reason: p = 2 > 1, so Σ 1/nᵖ converges to the exact value π²/6. partialSums: - 'n': '1' an: '1' sn-partial-sum: '1' - 'n': '2' an: '0.25' sn-partial-sum: '1.25' - 'n': '3' an: '0.111111' sn-partial-sum: '1.36111' - 'n': '4' an: '0.0625' sn-partial-sum: '1.42361' - 'n': '5' an: '0.04' sn-partial-sum: '1.46361' steps: >- Series: p-series (Σ 1/nᵖ) Convergence: Converges Reason: p = 2 > 1, so Σ 1/nᵖ converges to the exact value π²/6. Formula: S = ζ(p) (Riemann zeta function) Exact sum: π²/6 Numeric sum: 1.64493 Partial sums: S(1) = 1 (aₙ = 1) S(2) = 1.25 (aₙ = 0.25) S(3) = 1.36111 (aₙ = 0.111111) S(4) = 1.42361 (aₙ = 0.0625) S(5) = 1.46361 (aₙ = 0.04) credits_used: 5 credits_remaining: null leibniz_series_for_very_slow_convergence: summary: Leibniz series for π (very slow convergence) value: tool: infinite-series-sum-calculator tool_version: 1.0.2 outputs: converges: Converges exactSum: π/4 numericSum: '0.785398' formula: S = π/4 (from arctan(1) = π/4) reason: >- The Alternating Series Test applies (1/(2n+1) decreases to 0); this is the famous (very slowly converging) Leibniz formula for π. partialSums: - 'n': '0' an: '1' sn-partial-sum: '1' - 'n': '1' an: '-0.333333' sn-partial-sum: '0.666667' - 'n': '2' an: '0.2' sn-partial-sum: '0.866667' - 'n': '3' an: '-0.142857' sn-partial-sum: '0.72381' - 'n': '4' an: '0.111111' sn-partial-sum: '0.834921' steps: >- Series: Leibniz series for π (Σ (-1)ⁿ/(2n+1)) Convergence: Converges Reason: The Alternating Series Test applies (1/(2n+1) decreases to 0); this is the famous (very slowly converging) Leibniz formula for π. Formula: S = π/4 (from arctan(1) = π/4) Exact sum: π/4 Numeric sum: 0.785398 Partial sums: S(0) = 1 (aₙ = 1) S(1) = 0.666667 (aₙ = -0.333333) S(2) = 0.866667 (aₙ = 0.2) S(3) = 0.72381 (aₙ = -0.142857) S(4) = 0.834921 (aₙ = 0.111111) credits_used: 5 credits_remaining: null telescoping_series_k_2_n_1: summary: Telescoping series, k = 2, n₀ = 1 value: tool: infinite-series-sum-calculator tool_version: 1.0.2 outputs: converges: Converges exactSum: (1/2)·(1/1 + 1/2) numericSum: '0.75' formula: >- S = (1/k)·[1/n₀ + 1/(n₀+1) + ... + 1/(n₀+k-1)] (since 1/(n(n+k)) = (1/k)[1/n - 1/(n+k)] telescopes) reason: >- A telescoping series always converges — every interior term cancels, leaving only the first k partial fractions. partialSums: - 'n': '1' an: '0.333333' sn-partial-sum: '0.333333' - 'n': '2' an: '0.125' sn-partial-sum: '0.458333' - 'n': '3' an: '0.0666667' sn-partial-sum: '0.525' - 'n': '4' an: '0.0416667' sn-partial-sum: '0.566667' - 'n': '5' an: '0.0285714' sn-partial-sum: '0.595238' steps: >- Series: Telescoping series (Σ 1/(n(n+k))) Convergence: Converges Reason: A telescoping series always converges — every interior term cancels, leaving only the first k partial fractions. Formula: S = (1/k)·[1/n₀ + 1/(n₀+1) + ... + 1/(n₀+k-1)] (since 1/(n(n+k)) = (1/k)[1/n - 1/(n+k)] telescopes) Exact sum: (1/2)·(1/1 + 1/2) Numeric sum: 0.75 Partial sums: S(1) = 0.333333 (aₙ = 0.333333) S(2) = 0.458333 (aₙ = 0.125) S(3) = 0.525 (aₙ = 0.0666667) S(4) = 0.566667 (aₙ = 0.0416667) S(5) = 0.595238 (aₙ = 0.0285714) credits_used: 5 credits_remaining: null schema: type: object required: - tool - tool_version - outputs - credits_used - credits_remaining properties: outputs: type: object properties: converges: type: string description: Convergence exactSum: type: string description: Exact sum numericSum: type: string description: Numeric sum formula: type: string description: Formula reason: type: string description: Reason partialSums: type: array description: Partial sums items: type: object properties: 'n': type: string description: 'n' an: type: string description: aₙ sn-partial-sum: type: string description: Sₙ (partial sum) x-iotools-columns: - 'n' - an - sn-partial-sum steps: type: string description: Step-by-step derivation tool: type: string description: The tool's slug, echoing the {slug} in the request path. tool_version: type: string description: Output-contract version for this tool. credits_used: type: integer description: >- Credits this call consumed, after any settlement refund. 0 when metering is disabled. credits_remaining: type: - integer - 'null' description: >- Credits left in the current monthly allowance, or null when metering is disabled. request_id: type: string description: Correlation id, also sent as x-request-id. '400': $ref: '#/components/responses/ValidationError' '401': $ref: '#/components/responses/Unauthorized' '402': $ref: '#/components/responses/InsufficientCredits' '403': $ref: '#/components/responses/ToolNotAllowed' '404': $ref: '#/components/responses/ToolNotFound' '413': $ref: '#/components/responses/PayloadTooLarge' '422': $ref: '#/components/responses/UnprocessableInput' '429': $ref: '#/components/responses/RateLimited' '500': $ref: '#/components/responses/ToolFailed' '503': $ref: '#/components/responses/ToolDisabled' security: - bearerAuth: [] components: responses: ValidationError: description: Invalid request. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/validation_error title: type: string description: Short human-readable summary of the failure. examples: - Invalid request status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - validation_error detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. fields: type: - object - string additionalProperties: type: string description: >- What failed: a map of field name → message. Absent when the body itself is malformed; a single string when the failure isn't tied to one field. example: type: https://docs.iotools.cloud/errors/validation_error title: Invalid request status: 400 code: validation_error detail: One or more inputs are invalid — see `fields`. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 fields: inputString: Required Unauthorized: description: Missing or invalid API key. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/invalid_api_key title: type: string description: Short human-readable summary of the failure. examples: - Invalid API key status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - invalid_api_key detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/invalid_api_key title: Invalid API key status: 401 code: invalid_api_key detail: 'Provide ''Authorization: Bearer ''.' request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 InsufficientCredits: description: Monthly credit allowance exhausted. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/insufficient_credits - >- https://docs.iotools.cloud/errors/vendor_api_tier_limit_exceeded title: type: string description: Short human-readable summary of the failure. examples: - Insufficient credits - Free-tier API limit reached for this tool status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - insufficient_credits - vendor_api_tier_limit_exceeded detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: Always 0 — a refused call charges nothing. credits_remaining: type: integer description: Credits left in the allowance — fewer than this call costs. example: type: https://docs.iotools.cloud/errors/insufficient_credits title: Insufficient credits status: 402 code: insufficient_credits detail: This call costs 1 credit and 0 remain in this month's allowance. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 credits_remaining: 0 ToolNotAllowed: description: Tool exists but has no API surface. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_not_allowed title: type: string description: Short human-readable summary of the failure. examples: - Tool not available over the API status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_not_allowed detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_not_allowed title: Tool not available over the API status: 403 code: tool_not_allowed detail: >- "Background Remover" is available on iotools.cloud but has no API endpoint. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 ToolNotFound: description: No such tool. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_not_found title: type: string description: Short human-readable summary of the failure. examples: - Tool not found status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_not_found detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_not_found title: Tool not found status: 404 code: tool_not_found detail: No tool with that slug. See GET /v1/tools/list. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 PayloadTooLarge: description: Body too large. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/payload_too_large title: type: string description: Short human-readable summary of the failure. examples: - Payload too large status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - payload_too_large detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/payload_too_large title: Payload too large status: 413 code: payload_too_large detail: Request body exceeds this tool's size limit. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 UnprocessableInput: description: Well-formed request, but there is no result to return. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/unprocessable_input title: type: string description: Short human-readable summary of the failure. examples: - Input cannot be processed status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - unprocessable_input detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: >- Always 0 on `unprocessable_input` — a failed run is refunded, floor included. credits_remaining: type: integer description: >- Credits left after the refund. Absent when metering is disabled. example: type: https://docs.iotools.cloud/errors/unprocessable_input title: Input cannot be processed status: 422 code: unprocessable_input detail: >- This video has no captions or subtitles, so there is no transcript to extract. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 RateLimited: description: Per-minute rate limit exceeded. headers: Retry-After: description: Seconds to wait before retrying (RFC 9110 delta-seconds). schema: type: integer content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/rate_limited title: type: string description: Short human-readable summary of the failure. examples: - Rate limit exceeded status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - rate_limited detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. retry_after: type: integer description: >- Seconds until the window resets — the same value as the `Retry-After` header. example: type: https://docs.iotools.cloud/errors/rate_limited title: Rate limit exceeded status: 429 code: rate_limited detail: Too many requests. Retry in 30s. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 retry_after: 30 ToolFailed: description: Tool failed to run. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/processing_error - https://docs.iotools.cloud/errors/internal_error title: type: string description: Short human-readable summary of the failure. examples: - Tool failed to run - Internal error status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - processing_error - internal_error detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. credits_used: type: integer description: >- Always 0 on `processing_error` — a failed run is refunded, floor included. credits_remaining: type: integer description: >- Credits left after the refund. Absent when metering is disabled. example: type: https://docs.iotools.cloud/errors/processing_error title: Tool failed to run status: 500 code: processing_error detail: The tool failed to run. Please try again. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 credits_used: 0 ToolDisabled: description: Tool temporarily disabled. content: application/problem+json: schema: type: object description: >- RFC 9457 problem document, served as application/problem+json. Branch on `code`; `title` is human prose and may be reworded without notice. Some failures add extension members — `fields` on validation errors, `retry_after` on 429s, `credits_used`/`credits_remaining` on billing-adjacent failures — documented on the responses that carry them. required: - type - title - status - code properties: type: type: string format: uri description: Stable documentation URI for this failure. examples: - https://docs.iotools.cloud/errors/tool_disabled - https://docs.iotools.cloud/errors/api_unconfigured title: type: string description: Short human-readable summary of the failure. examples: - Tool temporarily disabled - API not configured status: type: integer description: HTTP status code, matching the response's own status. code: type: string description: >- Stable machine-readable error code — branch on this, not `title`. enum: - tool_disabled - api_unconfigured detail: type: string description: Human explanation of this occurrence. request_id: type: string description: Correlation id, also sent as x-request-id. example: type: https://docs.iotools.cloud/errors/tool_disabled title: Tool temporarily disabled status: 503 code: tool_disabled detail: This tool is temporarily unavailable. Try again shortly. request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11 securitySchemes: bearerAuth: type: http scheme: bearer bearerFormat: iot_live_… ````