infinite series

Approximating a function with a divergent series.

If you ignore exponentially small terms they can come back to bite you.

How a question about rainbows led to an entire field of mathematics.

Scientists have calculated the decimal digits of the number pi to a record 62.8 trillion places.

This problem about an infinite sum has a surprising answer.

How to write a sum that's infinitely long.

What is 1-1+1-1+1-1+...? How infinite sums challenge our notion of arithmetic.

What do you get when you add up all the natural numbers 1+2+3+4+ ... ? Not -1/12! We explore a strange result that has been making the rounds recently.

The number pi can be expressed beautifully in terms of infinite sums. For practical purposes though, these sums are rather disappointing: they converge slowly, so you need to sum a large number of terms to get accurate estimates of pi. Here's a clever way to make them converge faster.