Problem 82
Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{4 n^{2}-1} $$
Problem 82
Find all values of \(x\) for which the series converges. For these values of \(x,\) write the sum of the series as a function of \(x\). $$ \sum_{n=1}^{\infty}\left(\frac{x^{2}}{x^{2}+4}\right)^{n} $$
Problem 82
Give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.) A monotonically increasing bounded sequence that does not converge
Problem 82
State the Integral Test and give an example of its use.
Problem 83
Define a \(p\) -series and state the requirements for its convergence.
Problem 83
Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=1}^{\infty} \frac{\cos n}{2^{n}} $$
Problem 83
Give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.) A sequence that converges to \(\frac{3}{4}\)
Problem 83
(a) You delete a finite number of terms from a divergent series. Will the new series still diverge? Explain your reasoning. (b) You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.
Problem 84
Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=2}^{\infty} \frac{(-1)^{n}}{n \ln n} $$
Problem 84
Consider the formula \(\frac{1}{x-1}=1+x+x^{2}+x^{3}+\cdots\) Given \(x=-1\) and \(x=2\), can you conclude that either of the following statements is true? Explain your reasoning. (a) \(\frac{1}{2}=1-1+1-1+\cdots\) (b) \(-1=1+2+4+8+\cdots\)
