Based on what I read in a very old book, we show

We can divide the sum into even and odd contributions

One sees that

In order to evaluate the right hand side of this equation we consider the function

, such that

We now take a small detour and consider the finite sum and its derivatives

Multiplying the last equation by and using

We get

Now one can solve for

By writing

One can think of integration by parts and notice that the second term vanishes when . Now

In other words,

from which it is immediate

Now note and therefore

Finally, the identity establishes the result .

Happy new year to evri1!

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## About goobypl5

pizza baker, autodidact, particle physicist