power series calculator

Choose function / series

Center a

Evaluate at x

Number of terms (1-60)

Computes the partial sum S_n-1 = Σ from k=0 to n-1.

Enter values and click Calculate Partial Sum.

What this power series calculator does

This tool approximates a function by adding the first n terms of its Taylor (or related power) series around a center point a. In practical terms, it converts a possibly complicated function into a polynomial approximation and evaluates that polynomial at your chosen value of x.

The calculator returns the partial sum, exact function value (when defined), absolute error, convergence radius, and a term-by-term breakdown so you can see how the approximation evolves.

Quick refresher: what is a power series?

A power series centered at a has the form:

f(x) ≈ Σ c_k(x-a)^k, from k = 0 to n-1 for a partial sum.

For Taylor series, the coefficients are generated from derivatives: c_k = f^(k)(a) / k!.

Supported series in this calculator

e^x: converges for every real number.
sin(x) and cos(x): converges for every real number.
ln(1+x): converges for |x-a| < |1+a| (center must satisfy 1+a > 0).
1/(1-x): converges for |x-a| < |1-a| (center must satisfy a ≠ 1).

How to use it effectively

1) Pick a center near your evaluation point

In general, approximations improve when x is close to a. Large gaps often require many terms.

2) Increase terms gradually

Start with 5–8 terms, then increase to 12, 20, or more. Watch the absolute error shrink.

3) Check convergence status

For logarithmic and geometric-type series, being outside the radius usually means partial sums will not settle to the true function value.

Why this matters

Power series are foundational in numerical methods, differential equations, signal processing, and scientific computing. They provide a bridge between symbolic calculus and practical approximation, especially when exact algebraic manipulation is difficult.

Common mistakes to avoid

Using too few terms and assuming the approximation is accurate.
Ignoring convergence radius for ln(1+x) and 1/(1-x).
Choosing a center point very far from the target x.

Final tip

Treat the term table as a diagnostic tool. If terms are shrinking quickly and the running sum stabilizes, your approximation is usually in good shape.