Infinite Series Sum Calculator

Calculate the exact sum of convergent infinite series including geometric, telescoping, p-series, and well-known special series. Get step-by-step convergence proofs with animated partial sum visualizations.

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Geometric Series

$$\sum_{n=0}^{\infty} a \cdot r^n$$

Sum = a/(1−r) when |r| < 1. The most fundamental convergent series.

p-Series

$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$

Converges when p > 1. Includes the Basel problem (p=2) = π²/6.

Telescoping Series

$$\sum_{n=1}^{\infty} \frac{1}{n(n+k)}$$

Telescoping via partial fractions. For k=1: sum = 1.

Alternating Harmonic

$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln(2)$$

The alternating harmonic series. Sum = ln(2) ≈ 0.6931.

Leibniz Formula (π/4)

$$\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = \frac{\pi}{4}$$

The Leibniz formula. Sum = π/4 ≈ 0.7854. Slow convergence.

Basel Problem (π²/6)

$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

The Basel problem solved by Euler. Sum = π²/6 ≈ 1.6449.

Exponential (e)

$$\sum_{n=0}^{\infty} \frac{1}{n!} = e$$

The Taylor series for e. Converges extremely fast.

Geometric (Custom Start)

$$\sum_{n=N}^{\infty} a \cdot r^n$$

Geometric series with custom starting index N.

Alternating p-Series

$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^p}$$

Dirichlet eta function. For p=1 → ln(2), p=2 → π²/12.

Exponential eˣ

$$\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x$$

Taylor series for eˣ. Converges for all x.

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About Infinite Series Sum Calculator

The Infinite Series Sum Calculator computes the exact sum of convergent infinite series. It supports geometric series, p-series, telescoping series, and celebrated special series such as the Basel problem, the Leibniz formula for π, and the alternating harmonic series. Each calculation includes a step-by-step convergence proof, an animated partial sum visualization, and a detailed partial sums table.

Supported Series Types

∞

Geometric

a + ar + ar² + … = a/(1−r)

p-Series

Σ 1/nᵖ, converges if p > 1

⟿

Telescoping

Σ 1/n(n+k), partial fractions

Alternating

Σ (−1)ⁿ⁺¹/n = ln(2)

Leibniz

Σ (−1)ⁿ/(2n+1) = π/4

ℯ

Exponential

Σ 1/n! = e ≈ 2.718

Key Formulas

Series	Formula	Condition
Geometric	$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$	\|r\| < 1
p-Series	$\sum_{n=1}^{\infty} \frac{1}{n^p} = \zeta(p)$	p > 1
Telescoping	$\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1$	Always converges
Basel Problem	$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$	p-series with p = 2
Leibniz	$\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = \frac{\pi}{4}$	Alternating series
Alt. Harmonic	$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln(2)$	Conditional convergence
Exponential	$\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x$	All x ∈ ℝ

How to Use the Infinite Series Sum Calculator

Choose a series type: Click on a series card to select it, or use the quick example buttons for popular series. Use the category tabs to filter between Classic and Special series.
Enter parameters: If the series requires parameters (like the common ratio r for geometric series or the exponent p for p-series), fill in the input fields. Default values are provided.
Click Calculate Sum: Press the purple "Calculate Sum" button to compute the result.
Review the result: See the exact sum value, the animated partial sum convergence graph, the step-by-step mathematical proof, and the detailed partial sums table.

Understanding Convergence

An infinite series $\sum_{n=1}^{\infty} a_n$ converges if the sequence of partial sums $S_N = \sum_{n=1}^{N} a_n$ approaches a finite limit as N → ∞. The animated graph in our calculator shows this convergence visually — you can watch the partial sums approach the dashed limit line.

Key convergence tests:

Geometric Series Test: Σ arⁿ converges if and only if |r| < 1
p-Series Test: Σ 1/nᵖ converges if and only if p > 1
Alternating Series Test (Leibniz): Σ (−1)ⁿbₙ converges if bₙ is decreasing and approaches 0
Ratio Test: If lim|aₙ₊₁/aₙ| < 1, the series converges absolutely
Integral Test: Compare the series with an improper integral

Famous Results in Series Summation

Several infinite series have surprising and beautiful exact sums:

Basel Problem (1734): Euler proved that 1 + 1/4 + 1/9 + 1/16 + … = π²/6, connecting the sum of reciprocal squares to π.
Leibniz Formula (1674): The alternating series 1 − 1/3 + 1/5 − 1/7 + … = π/4, one of the simplest expressions for π.
Euler's Number: The series 1 + 1 + 1/2 + 1/6 + 1/24 + … = e ≈ 2.71828, converging extremely rapidly.
Alternating Harmonic Series: 1 − 1/2 + 1/3 − 1/4 + … = ln(2), despite the harmonic series itself diverging.

Frequently Asked Questions (FAQ)

What is an infinite series sum?

An infinite series sum is the result of adding infinitely many terms in a sequence. If the partial sums approach a finite number, the series is said to converge, and that number is its sum. For example, 1 + 1/2 + 1/4 + 1/8 + … = 2 is a convergent geometric series.

When does an infinite series converge?

An infinite series converges when its partial sums approach a finite limit. Different tests determine convergence: the Ratio Test, Root Test, p-Series Test, Alternating Series Test, and more. A necessary (but not sufficient) condition is that the terms must approach zero — the harmonic series 1 + 1/2 + 1/3 + … diverges even though terms approach zero.

What is the sum of a geometric series?

The sum of an infinite geometric series a + ar + ar² + … equals a/(1−r) when the absolute value of the common ratio r is less than 1. If |r| ≥ 1, the series diverges. For example, 1 + 1/2 + 1/4 + … = 1/(1−0.5) = 2.

What is the Basel problem?

The Basel problem asks for the exact sum of the reciprocals of the squares: 1 + 1/4 + 1/9 + 1/16 + … Euler solved it in 1734, proving the sum equals π²/6 (approximately 1.6449). This is one of the most celebrated results in number theory and analysis.

What is a telescoping series?

A telescoping series is one where consecutive terms cancel each other out, leaving only a finite number of terms in the partial sum. For example, the series Σ 1/(n(n+1)) can be written as 1/n − 1/(n+1) using partial fractions, and most terms cancel, giving a sum of 1.

Reference this content, page, or tool as:

"Infinite Series Sum Calculator" at https://MiniWebtool.com/infinite-series-sum-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-06

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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