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# Conclude if the given series converges or diverges and justify the conclusion

Test the following series for convergence or divergence. Justify the decision.

Let

Then, the series converges by Example #1 on page 398 of Apostol,

where (since ). Then consider the limit,

The limits of each of the terms in the product exist (as we show below) so the limit of the product is the product of the limits,

The limit since we know

for all , . Therefore, we have

By Theorem 10.9 (see the note after the proof of the theorem on page 396 of Apostol), we then have the convergence of implies the convergence of . Hence,

converges.

1. MathlessRick says:

Well as n -> infinity
Sqrt(2n-1) ~ sqrt(2n)
n(n+1) ~ n^2
log(4n+1) ~ log(4n) = log(4) + log(n)

So just solve for
ln4 * Sqrt(2n)/n^2 + sqrt(2)*ln(n)/n^(3/2)

Which clearly converges for both parts -> 0

So the sum must converge by theorem 10.10… right?