B Sury is with the Indian Statistical Institute. He introduces this article by:
Bernoulli truly stunned us with his number;
woke us up from a deep and ignorant slumber.
Its relation with Riemann zeta
makes us think nothing could be neater.
The connection is much deeper
– ask any plumber!

It is a beautiful discovery due to J.Bernoulli that for
any positive integer k, the sum sum_{i=1}^n i^k can be evaluated
in terms of, what are now known as, Bernoulli numbers.

In this article, we shall discuss several methods of evaluating
the above sum. For instance, Marikkannan and Ravichandran have
written about a method of evaluation using integration.
Apart from Bernoulli's method which we shall recall,
we give a method akin to using integration, and one using
differentiation. These methods are often useful in evaluating more
general sums too as we shall indicate. Finally, we discuss the connections with the Riemann Zeta function.

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Address for Correspondence

B Sury
Stat-Math Unit
Indian Statistical Institute
8th Mile Mysore Road
Bangalore 560 059, India.
Email: sury@isibang.ac.in