> ## 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. # Maclaurin Series Calculator API > Expand a common function — eˣ, sin(x), cos(x), sinh(x), cosh(x), ln(1+x), 1/(1±x), arctan(x), √(1+x), or arcsin(x) — into its Maclaurin series with exact rational coefficients, its radius of convergence, and an optional evaluation against the actual function value. [Try Maclaurin Series Calculator in your browser →](https://iotools.cloud/tool/maclaurin-series-calculator/) ## OpenAPI ````yaml https://api.iotools.cloud/v1/openapi post /v1/tool/maclaurin-series-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/maclaurin-series-calculator: post: tags: - Calculators summary: Maclaurin Series Calculator description: >- Expand a common function — eˣ, sin(x), cos(x), sinh(x), cosh(x), ln(1+x), 1/(1±x), arctan(x), √(1+x), or arcsin(x) — into its Maclaurin series with exact rational coefficients, its radius of convergence, and an optional evaluation against the actual function value. [Try Maclaurin Series Calculator in your browser →](https://iotools.cloud/tool/maclaurin-series-calculator/) operationId: run_maclaurin_series_calculator requestBody: required: false content: application/json: schema: type: object properties: func: enum: - exp - sin - cos - sinh - cosh - ln1px - geo - geo_alt - arctan - sqrt1px - arcsin description: Function f(x) nTerms: type: number description: Terms to show minimum: 3 maximum: 12 x0: type: number description: Evaluate at x required: [] examples: e_6_terms_evaluated_at_x_1: summary: eˣ, 6 terms, evaluated at x = 1 value: func: exp nTerms: '6' x0: '1' sin_x_4_terms_symbolic_only: summary: sin(x), 4 terms, symbolic only value: func: sin nTerms: '4' x0: '' arcsin_x_3_terms_evaluated_at_x_0_5: summary: arcsin(x), 3 terms, evaluated at x = 0.5 value: func: arcsin nTerms: '3' x0: '0.5' 1_x_4_terms_symbolic_only: summary: √(1 + x), 4 terms, symbolic only value: func: sqrt1px nTerms: '4' x0: '' responses: '200': description: Tool output content: application/json: examples: e_6_terms_evaluated_at_x_1: summary: eˣ, 6 terms, evaluated at x = 1 value: tool: maclaurin-series-calculator tool_version: 1.0.2 outputs: series: 1 + x + 1/2·x^2 + 1/6·x^3 + 1/24·x^4 + 1/120·x^5 coefficients: - 'n': '0' coefficient-exact-fraction: '1' term: '1' - 'n': '1' coefficient-exact-fraction: '1' term: x - 'n': '2' coefficient-exact-fraction: 1/2 term: 1/2·x^2 - 'n': '3' coefficient-exact-fraction: 1/6 term: 1/6·x^3 - 'n': '4' coefficient-exact-fraction: 1/24 term: 1/24·x^4 - 'n': '5' coefficient-exact-fraction: 1/120 term: 1/120·x^5 radius: ∞ (Converges for every real x.) approxValue: '2.71667' actualValue: '2.71828' error: '0.00161516' steps: >- f(x) = eˣ General term: aₙ = xⁿ / n! Radius of convergence: ∞ (Converges for every real x.) n = 0: coefficient = 1 n = 1: coefficient = 1 n = 2: coefficient = 1/2 n = 3: coefficient = 1/6 n = 4: coefficient = 1/24 n = 5: coefficient = 1/120 Maclaurin series: P(x) = 1 + x + 1/2·x^2 + 1/6·x^3 + 1/24·x^4 + 1/120·x^5 At x = 1: P(x) ≈ 2.71667, actual f(x) = 2.71828, |error| = 0.00161516 credits_used: 5 credits_remaining: null sin_x_4_terms_symbolic_only: summary: sin(x), 4 terms, symbolic only value: tool: maclaurin-series-calculator tool_version: 1.0.2 outputs: series: x - 1/6·x^3 + 1/120·x^5 - 1/5040·x^7 coefficients: - 'n': '1' coefficient-exact-fraction: '1' term: x - 'n': '3' coefficient-exact-fraction: '-1/6' term: 1/6·x^3 - 'n': '5' coefficient-exact-fraction: 1/120 term: 1/120·x^5 - 'n': '7' coefficient-exact-fraction: '-1/5040' term: 1/5040·x^7 radius: ∞ (Converges for every real x.) approxValue: '' actualValue: '' error: '' steps: >- f(x) = sin(x) General term: a₍₂ₖ₊₁₎ = (-1)ᵏ·x²ᵏ⁺¹/(2k+1)! (even-power terms are 0) Radius of convergence: ∞ (Converges for every real x.) n = 1: coefficient = 1 n = 3: coefficient = -1/6 n = 5: coefficient = 1/120 n = 7: coefficient = -1/5040 Maclaurin series: P(x) = x - 1/6·x^3 + 1/120·x^5 - 1/5040·x^7 credits_used: 5 credits_remaining: null arcsin_x_3_terms_evaluated_at_x_0_5: summary: arcsin(x), 3 terms, evaluated at x = 0.5 value: tool: maclaurin-series-calculator tool_version: 1.0.2 outputs: series: x + 1/6·x^3 + 3/40·x^5 coefficients: - 'n': '1' coefficient-exact-fraction: '1' term: x - 'n': '3' coefficient-exact-fraction: 1/6 term: 1/6·x^3 - 'n': '5' coefficient-exact-fraction: 3/40 term: 3/40·x^5 radius: 1 (Converges for -1 ≤ x ≤ 1 (including both endpoints).) approxValue: '0.523177' actualValue: '0.523599' error: '0.000421692' steps: >- f(x) = arcsin(x) General term: a₍₂ₖ₊₁₎ = (2k)! / (4ᵏ(k!)²(2k+1))·x²ᵏ⁺¹ (even-power terms are 0) Radius of convergence: 1 (Converges for -1 ≤ x ≤ 1 (including both endpoints).) n = 1: coefficient = 1 n = 3: coefficient = 1/6 n = 5: coefficient = 3/40 Maclaurin series: P(x) = x + 1/6·x^3 + 3/40·x^5 At x = 0.5: P(x) ≈ 0.523177, actual f(x) = 0.523599, |error| = 0.000421692 credits_used: 5 credits_remaining: null 1_x_4_terms_symbolic_only: summary: √(1 + x), 4 terms, symbolic only value: tool: maclaurin-series-calculator tool_version: 1.0.2 outputs: series: 1 + 1/2·x - 1/8·x^2 + 1/16·x^3 coefficients: - 'n': '0' coefficient-exact-fraction: '1' term: '1' - 'n': '1' coefficient-exact-fraction: 1/2 term: 1/2·x - 'n': '2' coefficient-exact-fraction: '-1/8' term: 1/8·x^2 - 'n': '3' coefficient-exact-fraction: 1/16 term: 1/16·x^3 radius: 1 (Converges for -1 < x ≤ 1 (diverges at x = -1).) approxValue: '' actualValue: '' error: '' steps: >- f(x) = √(1 + x) General term: aₙ = C(1/2, n)·xⁿ (generalized binomial coefficient) Radius of convergence: 1 (Converges for -1 < x ≤ 1 (diverges at x = -1).) n = 0: coefficient = 1 n = 1: coefficient = 1/2 n = 2: coefficient = -1/8 n = 3: coefficient = 1/16 Maclaurin series: P(x) = 1 + 1/2·x - 1/8·x^2 + 1/16·x^3 credits_used: 5 credits_remaining: null schema: type: object required: - tool - tool_version - outputs - credits_used - credits_remaining properties: outputs: type: object properties: series: type: string description: Maclaurin Series coefficients: type: array description: Coefficients items: type: object properties: 'n': type: string description: 'n' coefficient-exact-fraction: type: string description: Coefficient (exact fraction) term: type: string description: Term x-iotools-columns: - 'n' - coefficient-exact-fraction - term radius: type: string description: Radius of Convergence approxValue: type: string description: P(x) Approximation actualValue: type: string description: Actual f(x) error: type: string description: '|Error|' 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_… ````