Chapter 4

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. $$ a_{n}=\ln \left(\frac{1}{n}\right) $$

The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, \(\pi=3.141592653589793 .\)[T] The Euler transform rewrites \(S=\sum_{n=0}^{\infty}(-1)^{n} b_{n}\) as \(S=\sum_{n=0}^{\infty}(-1)^{n} 2^{-n-1} \sum_{m=0}^{n}\left(\begin{array}{c}n \\ m\end{array}\right) b_{n-m}\). For the alternating harmonic series, it takes the form \(\ln (2)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}=\sum_{n=1}^{\infty} \frac{1}{n 2^{n}} .\) Compute partial sums of \(\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}\) until they approximate \(\ln (2)\) accurate to within \(0.0001\). How many terms are needed? Compare this answer to the number of terms of the alternating harmonic series are needed to estimate \(\ln (2)\).

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. $$ a_{n}=\frac{\ln (n+1)}{\sqrt{n+1}} $$

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. $$ a_{n}=\frac{2^{n+1}}{5^{n}} $$

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. $$ a_{n}=\frac{\ln (\cos n)}{n} $$

Is the series convergent or divergent? $$ \sum_{n=1}^{\infty} \frac{1}{n^{2}+5 n+4} $$

Is the series convergent or divergent? $$ \sum_{n=1}^{\infty} \ln \left(\frac{n+1}{n}\right) $$

Is the series convergent or divergent? $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n^{4}} $$

Is the series convergent or divergent? $$ \sum_{n=1}^{\infty} \frac{e^{n}}{n !} $$

Is the series convergent or divergent? $$ \sum_{n=1}^{\infty} n^{-(n+1 / n)} $$