Taylor / Maclaurin Series Generator mechanism

Model scope

The core Taylor-polynomial method is standard after correcting the summation notation and resolving whether n denotes order or number of terms. The instrument cannot honestly claim a remainder estimate until it specifies the estimate method and its assumptions. It should label the result as an order-N Taylor polynomial, expose the convergence/domain limitations, and disclose any unsupported symbolic-function cases.

• The displayed formula is malformed: the summation index n is also used as the requested order, and the x-dependence is missing. The canonical order-N Taylor polynomial is P_N(x)=Σ_{k=0}^{N} f^(k)(a)(x-a)^k/k!. If the product requirement literally means N terms, the upper index should instead be N-1.
• The specification says 'to n terms' but the input says 'order n'; these differ by one term because order N normally includes derivatives k=0 through k=N, giving N+1 terms.
• A remainder estimate is claimed, but no remainder formula, bound assumptions, or error method is specified. A standard choice is the Lagrange bound |R_N(x)|≤M|x-a|^(N+1)/(N+1)!, requiring a valid bound M on |f^(N+1)| over the interval between a and x.
• The plotted comparison requires a stated domain and handling for points where f is undefined or where the Taylor series does not converge. The polynomial can be evaluated outside the function's domain or its convergence radius, but that comparison must be labeled honestly.

Units and conventions

• All inputs are dimensionless mathematical real numbers unless the function explicitly defines units; x and center a must use the same units.
• For trigonometric functions, angles are assumed to be in radians.
• The test vectors interpret the input as order N, using k=0 through k=N. This interpretation must be made explicit in the product.
• For ln(x), the real-valued domain is x>0; its Taylor series about a=1 converges for |x-1|<1, with endpoint behavior requiring separate handling.
• For 1/(1-x), the Taylor series about 0 has radius of convergence 1 because of the singularity at x=1.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "function": "exp(x)",
    "center_a": "0",
    "order_N": 3,
    "evaluation_points": [
      "0.5000000000000000",
      "-1.000000000000000"
    ]
  },
  "expected_outputs": {
    "canonical_exact_symbolic_result": "P_3(x)=1+x+x^2/2+x^3/6",
    "spot_checks": [
      {
        "at": "x=0.5000000000000000",
        "input_side": 1.6487212707001282,
        "answer_side": 1.6458333333333333
      },
      {
        "at": "x=-1.000000000000000",
        "input_side": 0.3678794411714423,
        "answer_side": 0.3333333333333333
      }
    ],
    "remainder_estimate": "Not determinable from the specification because no remainder method or derivative bound is supplied."
  },
  "reference_data": {
    "interpretation": "order_N means terms k=0 through k=N",
    "exact_function_values": {
      "exp(1/2)": "e^(1/2)",
      "exp(-1)": "e^(-1)"
    },
    "exact_polynomial_values": {
      "P_3(1/2)": "79/48",
      "P_3(-1)": "1/3"
    }
  },
  "working": "Derivatives of exp(x) are all exp(x), so at a=0 every derivative equals 1. Therefore P_3(x)=1/0!+x/1!+x^2/2!+x^3/3!=1+x+x^2/2+x^3/6. At x=0.5: 1+0.5+(0.5^2)/2+(0.5^3)/6=1+0.5+0.125+0.0208333333333333=1.6458333333333333. The original value is exp(0.5)=1.6487212707001282. At x=-1: 1-1+1/2-1/6=1/3=0.3333333333333333. The original value is exp(-1)=0.3678794411714423."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The generator must reject centers where the function or required derivatives are undefined, must detect nondifferentiable branch switches, and must require an explicit angle-unit convention. Otherwise it can emit undefined coefficients or materially incorrect expansions.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche8/tranche8-results.json
• BOUNDARY_HAZARD_CATALOG.json