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,

So
,
and thus
.
And we also can get:
.

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