Solutions to the book Amazing and Aesthetic Aspects of Analysis by Paul Loya
Solutions to the book Galois Theory by Ian Stewart
Solutions to the book Complex Variables by M. Ya. Antimirov, A. A. Kolyshkin and Remi Vaillancourt
Solutions to the book Introduction to Analytic Number Theory by Tom M. Apostol
Sums of the form where , sometimes are called Euler sums. There are several ways to evaluate these sums. Aabout 240 years ago, Leonhard Euler(1707-1783) in 1775 proved that for ,
In the following you can see an elementary proof of this formula in the three steps:
Step 1:
Hence,
Step 2:
Now by recent formula that is also an alternate definition of the Harmonic numbers, we can write:
Step 3: