Problem: Investigate convergence of the following series. (Does it converge? If it does, then how?)
Solution: Note that all the terms are positive, so convergence and absolute convergence are the same for it. Thus it is remains to check on convergence and we have all the nice tests for series with positive terms available.
Since the terms of the series consist of powers, the natural candidate would be the Ratio test. Due to the dual nature of terms we will explore what the relevant ratios are doing separately for even and odd k's. Using l'Hôpital's rule we get
We see that the sequence
How about the Root test? Again, we first look at what the even and odd terms are doing.
So again we have no limit. This time we have a finite limes superior, but it
is equal to 1 and thus it does not help. How about the general version with
inequalities? Note that all terms
What other test could help? Since the terms are not given by one formula, the Integral test is clearly out. Our only hope now lies in some comparison. The results above suggest that there is no limit comparison possible, since the even and the odd terms behave in entirely different ways; the scale of powers shows that the even terms go to zero incomparably faster than the odd terms.
Thus the only hope remains with comparison. Actually, the last sentence of the above paragraph suggests one possible approach. It is relatively simple to check that 2k is greater than k2 for k at least 5 (or we just appeal to the above-mentioned scale of powers and conclude that such inequality must be true for large k), therefore we get comparison
Since the series on the right converges (see the p-test or the Example there, this series is quite well-known), it follows that also the given series converges, therefore also converges absolutely.
Alternative: There is another interesting trick that one could
try. The given series has terms of two kinds, thus we can try to separate
them into two independent series. We therefore express it as the sum of the
series
What can we say about convergence of these two series? At the first glance we did not really improve our situation, both of them are of dual nature again, but - and that is the trick - zeros can be dropped from a series without any trouble. Thus we can rewrite these series using the usual expressions for even and odd numbers as follows.
We see that the first series is a
geometric series with
The sum of two convergent series again converges, which confirms the convergence of the given series.