Problem: Determine whether the following series converge.

Solution: We start with the first series. Its form invites us to use the Root test.

Since ϱ < 1, the first given series converges. Note that no other tests are of any help here with exception of the Ratio test, but the appropriate limit is rather hard to work out.

Now we pass to the second given series. Is the Root test so nice again? This time we get an indeterminate power and we have to use the appropriate trick, see Sequences - Methods Survey - Limit.

So in this case the Root test does not help, obviously the Ratio test would be also inconclusive and so we have to look elsewhere for help. While the function f (x) = 21/x − 1 is decreasing and positive, we would not know how to integrate it and so the Integral test is out.

How is it with comparison tests? We could try the obvious comparison

but since the large series on the right diverges (its terms do not go to 0), it is useless. What else can we do? Evaluating ratio of successive terms looks like a very bad business, so it is not a good idea to try even more complicated refinements of the Ratio test (like Raabe test etc.). We may try another angle: What do we know of ak when k is large? Then 1/k is very small and positive, and we can try to approximate 2y around the origin by its tangent line there to get a nicer expression. We have

so we may guess that for large values of k we have

If we want to use the Limit comparison test, we have to justify such a claim.

Now we are justified to compare

Since the test series on the right diverges (it is the famous harmonic series, or see the p-test), it follows that also the given series diverges.


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