91Ó°ÊÓ

Suppose that \(\sum_{k=1}^{\infty} a_{k}\) is a convergent series of positive terms. Show that \(\sum_{k=1}^{\infty_{0}} a_{k}^{2}\) is convergent. Does the converse hold?

Prove this more general version of the Cauchy condensation test: If the terms of a series \(\sum_{k=1}^{\infty} a_{k}\) are nonnegative and decrease monotonically to zero, then that series converges if and only if the related series $$ \sum_{j=1}^{\infty}\left(m_{j+1}-m_{j}\right) a_{m_{j}} $$ converges. Here \(m_{1}

Obtain a proof that every series \(\sum_{k=1}^{\infty} a_{k}\) for which \(\sum_{k=1}^{\infty}\left|a_{k}\right|\) converges must itself be convergent without using the Cauchy criterion.

Express the infinite repeating decimal $$ .123451234512345123451234512345 \ldots $$ as the sum of a convergent geometric series and compute its sum (as a rational number) in this way.

Suppose that a bird flying 100 miles per hour (mph) travels back and forth between a train and the railway station, where the train and the bird start off together 1 mile away and the train is approaching the station at a fixed rate of \(60 \mathrm{mph}\). How far has the bird traveled when the train arrives? You most likely did not use a geometric series; can you find an argument that does?

Determine for what values of \(p\) the series $$ \sum_{k=1}^{\infty}(-1)^{k-1} \frac{1}{k^{p}}=1-\frac{1}{2^{p}}+\frac{1}{3^{p}}-\frac{1}{4^{p}} \ldots $$ is absolutely convergent and for what values it is nonabsolutely convergent.