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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Tags: Infinite Series, integrals, Math, Monday Math

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