Sum of Series Calculator

What Is a Series Sum?

A series is the sum of the terms of a sequence. This calculator handles arithmetic series (constant difference between terms) and geometric series (constant ratio between terms), including infinite geometric sums when they converge.

Arithmetic Series Partial Sum

For an arithmetic sequence with first term $a$, common difference $d$, and $n$ terms:

$$S_n = \frac{n}{2}\left[2a + d(n-1)\right] = \frac{n}{2}(a + a_n)$$

Geometric Series Partial Sum

For a geometric sequence with first term $a$, ratio $r$, and $n$ terms (when $r \neq 1$):

$$S_n = \frac{a(1 - r^n)}{1 - r}$$

Infinite Geometric Series

When $|r| < 1$, the infinite sum converges to:

$$S = \frac{a}{1 - r}$$

If $|r| \geq 1$, the series diverges and has no finite sum.

Related: Sum of Linear Number Sequence Calculator and Geometric Sequence Generator.