Mathematics

Problem

Four series are listed below. A.

\displaystyle \sum_{n=1}^{\infty} (-1)^{n}\frac{n}{n+1}

\displaystyle \sum_{n=1}^{\infty} \frac{1}{n!}

\displaystyle \sum_{n=1}^{\infty} \frac{\cos n}{\sqrt{n}}

\displaystyle \sum_{n=1}^{\infty} \frac{2^{n}}{n^{2}}

(a) Compute

\displaystyle \lim_{n \to \infty} a_n

for each series. (b) Rank the four series according to the decisiveness of the limit test as follows: • Category I: Diverges by the limit test • Category II: Converges by the limit test • Category III: Limit test inconclusive Explain the placement of each series in your ranking. (c) Choose one series from Category III. Propose and outline an additional convergence test that could be applied to that series, making clear how it would overcome the limitation of the limit test.

Browse the problem library