Problem: Determine whether the following series converges.

Solution: The terms of this series are positive, so we may use all the nice tests, but perhaps we should start by looking closer at these terms. We see that ak = 1/2k for k even and ak = 1/4k for k odd. Thus we do not have a nice expression (we cannot express it as a function) and so we cannot hope to use the Integral test.

For series featuring powers we often use the Ratio test. How does the relevant ratio look like?

When k grows to infinity, then the odd ratios run away to infinity, while the even ones go to zero. This shows that the limit for λ does not exist and the Ratio test does not work in its limit version. The dual behavior shows that even the more general versions of Ratio test (those with inequalities) do not work, since we cannot force the ratio ak+1/ak to stay below 1, nor can we force it to stay above 1. Thus this test is no good here.

How about the Root test?

Just like with the Ratio test, the limit for ϱ does not exist and so the limit version of the Root test does not work. However, here we have a chance to use the more general version with inequalities. Indeed, if we take q = 1/2, then q < 1 and for all k we have

This proves that the given series is convergent.

Alternative: We could also try to use comparison. Due to the schizophrenic nature of this series we cannot hope to say that the terms look like a certain something for k large, which rules out the Limit comparison test, but we actually have a nice opportunity to use the plain Comparison test. We have

Since the series on the right converges (it is a geometric series with |q| < 1), also the series on the left must converge.

Alternative: There is yet 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  ∑ ak  and  ∑ bk,  where

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 in both cases we have geometric series, both have |q| < 1, therefore they converge. The sum of two convergent series again converges, which confirms the convergence of the given series.


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