How to | Evaluate Infinite Sums and Products

In calculus, infinite sums and products can pose a challenge to manipulate by hand. The Wolfram Language can evaluate a huge number of different types of sums and products with ease.

Use Sum to set up the classic sum , with the function to sum over as the first argument. Use the Wolfram Language's usual range notation {variable,minimum,maximum} as the second argument:

This also works for finite sums like :

Use 1. to get the decimal representation:

This checks that :

Some functions have an infinite sum representation, and the Wolfram Language will recognize these. For example :


Many functions have product representations as well, and the Wolfram Language will even recognize these.

Use Product to check , a function found by the mathematician Euler. The arguments of Product have the same form as Sum:

Even more abstract functions will be recognized; the Product representation of involves the set of prime numbers: