What is a power series?

A power series is an infinite sum of the form Σ aₙ(x - c)ⁿ, where c is the center. When c = 0, it is called a Maclaurin series. Taylor series are power series centered at an arbitrary point c.

How many terms should I use?

For most functions near x = 0, 10–15 terms give excellent precision. Near the edge of the convergence radius (e.g. x = ±1 for ln(1+x)), you may need 50+ terms for good accuracy.

Why does the partial sum differ from the exact value?

A partial sum uses a finite number of terms. The difference between the partial sum and the true value is the truncation error, which decreases as you add more terms.

What is the radius of convergence?

It is the range of x values for which the series converges. For e^x, sin(x), and cos(x) it is infinite. For ln(1+x) it is |x| ≤ 1 (with x ≠ -1), and for (1+x)^α it depends on α.

Are these Maclaurin or Taylor series?

All series here are centered at c = 0, so they are Maclaurin series, which are a special case of Taylor series.

Power Series Calculator

Name: Power Series Calculator
Availability: InStock
Author: Nham Vu

Pick a function, enter x and the number of terms, and instantly see the partial sum with a full term-by-term breakdown.

Series Settings

Function

Value of x

Number of terms: 10

General term formula

eˣ = Σ xⁿ / n! (n = 0 to ∞)

Partial Sum (N terms)

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Exact Value

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Absolute Error

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Convergence Accuracy

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Fraction of exact value captured by the partial sum

Term-by-Term Breakdown

n	Term formula	Term value	Running sum
Press Calculate to see the term breakdown.

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Summary

Pick a function, enter x and the number of terms, and instantly see the partial sum with a full term-by-term breakdown.

How it works

Select a function from the dropdown (e.g. e^x, sin(x), cos(x)).
Enter the value of x where you want to evaluate the series.
Choose how many terms (N) to include in the partial sum.
The calculator computes each term using the series formula and sums them.
Results show the partial sum, the exact value, and the error.
The term table lists every term's index, formula, and numeric value.

Use cases

Verify hand-calculated Taylor series approximations for homework.
Explore how quickly a power series converges as more terms are added.
Check series values at a specific point for calculus or analysis courses.
Visualize the term-by-term breakdown of a Maclaurin expansion.
Confirm convergence radius intuition by testing large or small x values.
Quickly look up the general term formula for common series.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu