infinite series

Euler’s Product Formula

Here is an amazing formula due to Euler:SUMn=1 to infinity n-s = PRODp prime (1 – p-s)-1 .What’s interesting about this formula is that it relates an expression involving all the positive integers to one involving just primes! And you can use it to prove there must be infinitely many primes....

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Thinned-Out Harmonic Series

You’re probably already aware that the harmonic series, which is the sum of the reciprocals of all natural numbers, diverges. In fact, it diverges if you take away every other term. It even diverges if you take away nine out of every ten terms. So, what...

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Sums of Reciprocal Powers

You have seen that the harmonic series diverges. What about the sum of reciprocal squares? In fact, they converge, and to something very interesting: SUMk=1 to infinity ( 1/k2 ) = ( Pi2/6 ) Where did that Pi come from, anyway? If you liked that one, here are more:...

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