Taylor / Maclaurin Series Polynomial Approximation & Lagrange Remainder Bound Calculator

Evaluate N-th degree Maclaurin expansion of e^x and exact truncation remainder |R_N(x)|.

Degree-5 Maclaurin Approximation P_5(1.5)
4.462 ≈ e^1.5
Exact Value e^x 4.482 exact
Actual Truncation Error |e^x − P_n(x)| 0.02 error
Lagrange Remainder Upper Bound 0.071 bound
Inputs
⚡ Scenarios:
x
Value of x at which to evaluate the series.
degree n
Degree N of the Maclaurin polynomial (1 to 15).
Results & Graph
Degree-5 Maclaurin Approximation P_5(1.5)
4.462 ≈ e^1.5
Exact Value e^x
4.482 exact
Actual Truncation Error |e^x − P_n(x)|
0.02 error
Lagrange Remainder Upper Bound
0.071 bound

Degree-5 Maclaurin Approximation P_5(1.5)

Graph 1 • Taylor Series & Lagrange Error • Domain Architecture & Live Telemetry Maps Evaluation Point X, Polynomial Degree N through Taylor Series & Lagrange Error Formula.
Degree-5 Maclaurin Approximati: 4.462 ≈ e^1.5 Taylor Series & Lagrange Error • Invariants & Metrics Degree-5 Maclaurin Approxima 4.462 ≈ e^1.5 Exact Value e^x 4.482 exact Actual Truncation Error |e^x 0.02 error Lagrange Remainder Upper Bou 0.071 bound
Graph 2 • Degree-5 Maclaurin Approximation P_5(1.5) Across Evaluation Point X Sweep (40% → 160%) Evaluates P_n(x) = Σ_{k=0..n} (x^k / k!); LagrangeErrorBound |R_n(x)| ≤ max|f^(n across 7 operating points (current input highlighted).
0 ≈ e^1. 5.85 ≈ e^1. 11.69 ≈ e^1. 1.82 0.600x 2.46 0.900x 3.31 1.20x 4.46 1.50x 5.99 1.80x 8.00 2.10x 10.63 2.40x
Graph 3 • Joint Sensitivity: Evaluation Point X × Polynomial Degree N Compares Degree-5 Maclaurin Approximation P_5(1.5) when Polynomial Degree N is shifted +25% (solid) vs. −25% (dashed).
0 ≈ e^1. 3.85 ≈ e^1. 7.70 ≈ e^1. --- Polynomial Degree N @ 75% 2.86 -30% 3.58 -15% 4.48 +0% 5.60 +15% 7.00 +30%
Summary: Degree-5 Taylor polynomial gives P_5(1.5) = 4.46172 vs. exact e^1.5 = 4.48169 (true error 0.01997 ≤ Lagrange bound 0.0709).
Summary: Degree-5 Taylor polynomial gives P_5(1.5) = 4.46172 vs. exact e^1.5 = 4.48169 (true error 0.01997 ≤ Lagrange bound 0.0709).
⚡ Send result to: Scientific →Statistics & SD →Probability & P-Value →
Interactive chart & math engine powered by CalculatorGraph

Formula & Mathematical Methodology: Taylor Series & Lagrange Error Formula

Pn(x) = Σk=0..n (xk / k!)
LagrangeErrorBound |Rn(x)| ≤ max|fn+1(c)| × |x|^n+1n+1!
P_n(x) = Σ_{k=0..n} (x^k / k!); LagrangeErrorBound |R_n(x)| ≤ max|f^(n+1)(c)| × |x|^(n+1) / (n+1)!
  • evaluationPointX: Evaluation Point X (x)
  • polynomialDegreeN: Polynomial Degree N (degree n)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Taylor Series & Lagrange Error

Evaluate N-th degree Maclaurin expansion of e^x and exact truncation remainder |R_N(x)|.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: P_n(x) = Σ_{k=0..n} (x^k / k!); LagrangeErrorBound |R_n(x)| ≤ max|f^(n+1)(c)| × |x|^(n+1) / (n+1)!.
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Taylor Series & Lagrange Error via P_n(x) = Σ_{k=0..n} (x^k / k!); LagrangeErrorBound |R_n(x)| ≤ max|f^(n+1)(c)| × |x|^(n+1) / (n+1)!.

Frequently Asked Questions

How does the Taylor / Maclaurin Series Polynomial Approximation & Lagrange Remainder Bound Calculator compute the solution?

It evaluates P_n(x) = Σ_{k=0..n} (x^k / k!); LagrangeErrorBound |R_n(x)| ≤ max|f^(n+1)(c)| × |x|^(n+1) / (n+1)! directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Numerical Analysis & Calculus

Taylor / Maclaurin Series Polynomial Approximation & Lagrange Remainder Bound Calculator: inputs, method, and how to read the result

Evaluate N-th degree Maclaurin expansion of e^x and exact truncation remainder |R_N(x)|.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Evaluation Point X: Value of x at which to evaluate the series.
  • Polynomial Degree N: Degree N of the Maclaurin polynomial (1 to 15).

What the calculation includes

With the default values shown, Degree-5 Maclaurin Approximation P_5(1.5) is 4.462 ≈ e^1.5. The calculation follows Taylor Series & Lagrange Error Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Evaluation Point X from 1.5 x to 1.65 x changes Degree-5 Maclaurin Approximation P_5(1.5) (4.462 ≈ e^1.5) to Degree-5 Maclaurin Approximation P_5(1.65) (5.171 ≈ e^1.65). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

More in Mathematics & Statistics: Scientific Calculator, Statistics & Standard Deviation Calculator, Permutations, Combinations, Probability, Sample Size & P-Value Calculator.