Dear all, how can I prove the following summation (See attachment)?

I tried to use integrals but the result was kind of broad.

Attachment 29481

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- Oct 15th 2013, 07:10 AMmohamedennahdiProving a summation
Dear all, how can I prove the following summation (See attachment)?

I tried to use integrals but the result was kind of broad.

Attachment 29481 - Oct 15th 2013, 07:16 AMSlipEternalRe: Proving a summation

Assume the formula is correct for all nonnegative integers up to .

By the induction assumption, we have

Can you finish from here? - Oct 15th 2013, 07:23 AMSlipEternalRe: Proving a summation
Alternately, if you start with

And differentiate both sides with respect to , you can also get the formula you want. - Oct 15th 2013, 07:34 AMmohamedennahdiRe: Proving a summation
Thank you.

abd Could you help me with how we can come up with the formula from the summation?

As how to deduce the formula from the sigma notation? - Oct 15th 2013, 07:50 AMSlipEternalRe: Proving a summation
The original formula? Or the formula I wrote: ? Because that formula is just the standard geometric sum:

- Oct 15th 2013, 11:35 AMmohamedennahdiRe: Proving a summation
Imagine you have only the sigma notation, like this:

How would you deduce the following formula from the above summation?

- Oct 15th 2013, 12:11 PMSlipEternalRe: Proving a summation
I would think it looks like the derivative of a geometric sum. So, I would start with a geometric sum, and take the derivative to see what I get.