Problem: Find the sum of the series

Solution: This series does not look like something that can be transformed into a geometric series. It also definitely does not come from some known or simple power series. Indeed, the given series does not feature terms of the form c^k, so the only way to get it from a power series is to start with a power series that has the complicated fraction with roots as coefficient a_k and 1 is substituted into x^k. Frankly, I see no way to handle such a dreadful power series.

Thus there are two possibilities left. Either we find this sum by somehow transforming it into a telescopic series, or we show that it diverges and thus sidestep the necessity of summing it up. At first sight the telescopic angle does not look too inviting, so we try to check on convergence. The terms of the series are all positive (check), so we have a whole range of tests available. We definitely do not want to integrate the term in question, so the Integral test is out, and also the idea of taking the limit of the k-th root of terms sounds insane, so the Root test is out. Ratio also does not look too inviting, so it is up to comparison, which is no surprise as we have a ratio of polynomials and roots, exactly the right type for comparison. How does an individual term look like for k near infinity? We have a little problem here, since in the denominator the dominant term k appears twice and gets subtracted, which is exactly the situation when intuitive evaluation does not work. In such a case we have to get rid of the roots in the numerator algebraically.

We see that for large k the terms of the series essentially look like 1/(2k²). This should be confirmed by calculating the appropriate limit, see Limit comparison test in Methods Survey - Testing convergence, but there is no point in going through this. We were hoping to prove that the given series diverges, but it behaves just like the series of 1/k² that is known to converge (see the p-test in Theory - Introduction - Important examples or the appropriate Example there). Thus our given series seems to be convergent and we do have to find what its sum is.

We are down to our last hope, we need to change the series into one that fits into the telescopic series form. The form of the fraction suggests that it could be written as a difference, but we need to write the two resulting parts is such a way that they should come from the same pattern, just with different k. We try it:

We did get a telescopic series, so we apply the appropriate procedure, namely we check on it partial sums.

Since the partial sums converge, we can conclude that

Next problem
Back to Solved Problems - Summing up series