Problem 12 Sums Series

Problem 12: Sums & Series Problem: Evaluate the infinite sum

∞ X n=1

n4

HMMT 2008 Guts #14

n . +4

Solution: We can use the Sophie Germain identity, which states that

x4 + 4y 4 = (x2 + 2xy + 2y 2 )(x2 − 2xy + 2y 2 ) A B n in the form 2 + . Using Partial Fraction n4 + 4 n + 2n + 2 n2 − 2n + 2 −1 1 Decompostion, we find A = and B = . We can factor out the 14 and rewrite the sum: 4 4 Now we can rewrite

∞ X n=1

∞

n 1X −1 1 = + 2 2 2 n +4 4 n + 2n + 2 n − 2n + 2 n=1

If we start wrting out the terms, we notice almost everything telescopes, and we are left with   1 3 1 1+ = 4 2 8

Solution was written by Sean Soni and compiled from Art of Problem Solving Forums.