Series Sum Calculator

Math Algebra • Sequences and Series

Written by STEM Calculators Team Published December 19, 2025 Updated February 24, 2026

4. Series Sum Calculator

Compute finite/infinite sums for geometric, arithmetic, p-series (plus telescoping presets). Shows closed form when available, numeric partial sums, convergence/divergence, and a partial-sum plot.

Preset:

Series family: Sum type:

Parameters

First term a (at k=m):

Ratio r:

Start index m:

End index n (finite only):

Plot partial sums up to n (max 400):

Infinite-sum tolerance (numeric approx):

Max terms for numeric approx (∞):

Graph options:

Connect points Show values on points Auto label density Show limit line (if known)

Ready

Partial sum plot

x-axis: n (upper index) • y-axis: Sₙ (partial sum). Drag to pan • wheel/pinch to zoom.

n: 0, Sₙ: 0 sx: 60, sy: 35

Click Calculate to get the closed form (when available), convergence/divergence, and partial sums.

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Theory: Recognizing and Summing Common Series

A series is a sum of terms \(t_k\): \[ \sum_{k=m}^{n} t_k \quad\text{(finite)}, \qquad \sum_{k=m}^{\infty} t_k \quad\text{(infinite)}. \] The partial sums are \(S_n=\sum_{k=m}^{n} t_k\). An infinite series converges if \(S_n\) approaches a finite limit.

1) Geometric series

Geometric terms have a constant ratio: \(t_k=a\,r^{k-m}\) (so \(t_m=a\)).

\[ \sum_{k=m}^{n} a\,r^{k-m} = a\,(1+r+\cdots+r^{N-1}) = \begin{cases} a\,\dfrac{1-r^{N}}{1-r}, & r\neq 1,\\[8pt] a\,N, & r=1, \end{cases} \quad N=n-m+1. \]

The infinite geometric series converges exactly when \(|r|<1\):

\[ \sum_{k=m}^{\infty} a\,r^{k-m}=\frac{a}{1-r}\quad (|r|<1). \]

2) Arithmetic series

Arithmetic terms have constant difference: \(t_k=a+(k-m)d\).

\[ \sum_{k=m}^{n} \bigl(a+(k-m)d\bigr) = \frac{N}{2}\left(2a+(N-1)d\right), \quad N=n-m+1. \]

An infinite arithmetic series does not converge unless every term is 0 (because the terms do not approach 0).

3) p-series and the p-test

A p-series has the form \[ \sum_{k=m}^{\infty} \frac{A}{k^p}. \] The p-test says it converges if and only if \(p>1\) (for \(m\ge 1\)).

Many p-series do not have an elementary closed form, but partial sums can approximate the value. A standard bound (for decreasing positive terms) comes from the integral test:

\[ \left|R\right|=\left|\sum_{k=K+1}^{\infty}\frac{A}{k^p}\right| \le \frac{|A|\,(K+1)^{1-p}}{p-1}\quad (p>1). \]

A famous special case is the Basel series:

\[ \sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}. \]

4) Telescoping series (extension)

A series is telescoping if terms cancel after rewriting. A classic example:

\[ \frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1} \Rightarrow \sum_{k=m}^{n}\frac{1}{k(k+1)}=\frac{1}{m}-\frac{1}{n+1}. \]

Letting \(n\to\infty\) gives \(\sum_{k=m}^{\infty}\frac{1}{k(k+1)}=\frac{1}{m}\).

Frequently Asked Questions

How do I find the sum of a geometric series?

A geometric series has terms t_k = a r^(k-m). For a finite sum with N = n - m + 1, the sum is a(1 - r^N)/(1 - r) when r != 1; for an infinite sum it converges to a/(1 - r) only when |r| < 1.

What is the formula for the sum of an arithmetic series?

An arithmetic series has terms t_k = a + (k-m)d. With N = n - m + 1 terms, the finite sum is N/2 (2a + (N-1)d).

When does a p-series converge?

A p-series has the form sum_{k=m to infinity} A/k^p with m >= 1. It converges if p > 1 and diverges if p <= 1.

What makes a series telescoping in this calculator?

The telescoping preset uses terms of the form A/(k(k+1)), which can be rewritten as A(1/k - 1/(k+1)). This creates cancellations in partial sums and yields a simple closed form.

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Frequently Asked Questions

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