Monday Math 81

Recall from here that
Letting and solving for the sum, we see
Now, , so the above becomes

Now, along with our formula that
for positive integer n (see here), we can derive a couple of interesting results.

First, consider . Expanding the sine in its Maclaurin series:

Now, suppose we instead expanded the denominator of the integrand of I(x) via a geometric series as here:
Via multiple integration by parts or a table of integrals,

and thus
And we also can get:


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