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// from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-06

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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

Infinite Series Sum Calculator

About Infinite Series Sum Calculator

Supported Series Types

Key Formulas

How to Use the Infinite Series Sum Calculator

Understanding Convergence

Famous Results in Series Summation

Frequently Asked Questions (FAQ)

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