Summations arise regularly in mathematical analysis. For example,

Having a simple *closed form* expression such as makes the summation easier to understand and evaluate.

The summation we focus on in this post is

We will find a closed form for it.

In a recent post, I derived the closed form of a simpler summation (see “Beer theorems and their proofs“) Namely,

From (2) it follows that

which gives us

.

Or,

.

Therefore,

.

Let we arrived at (1)’s closed form:

.

I have a Computer Algebra aided solution too.

Let ,

we have

Therefore, the closed form of is the solution of initial-value problem

It is solved by Omega CAS Explorer (see Fig. 1)

Fig. 1

At ACA 2017 in Jerusalem, I gave a talk on “Generating Power Summation Formulas using a Computer Algebra System“.

I had a dream that night. In the dream, I was taking a test.

It reads:

*Derive the closed form for *

I woke up with a sweat.

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