Problem 77
Determine the convergence or divergence of the series. $$ \frac{1}{200}+\frac{1}{400}+\frac{1}{600}+\frac{1}{800}+\cdots \cdot $$
Problem 78
Let \(\left\\{a_{n}\right\\}\) be a monotonic sequence such that \(a_{n} \leq 1\). Discuss the convergence of \(\left\\{a_{n}\right\\} .\) If \(\left\\{a_{n}\right\\}\) converges, what can you conclude about its limit?
Problem 78
Determine the convergence or divergence of the series. $$ \frac{1}{200}+\frac{1}{210}+\frac{1}{220}+\frac{1}{230}+\cdots $$
Problem 78
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 \(x\). $$ \sum_{n=0}^{\infty} 4\left(\frac{x-3}{4}\right)^{n} $$
Problem 79
Determine the convergence or divergence of the series. $$ \frac{1}{201}+\frac{1}{204}+\frac{1}{209}+\frac{1}{216}+\cdots \cdot $$
Problem 79
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 \(x\). $$ \sum_{n=0}^{\infty}(-1)^{n} x^{n} $$
Problem 79
In your own words, define each of the following. (a) Sequence (b) Convergence of a sequence (c) Monotonic sequence (d) Bounded sequence
Problem 80
Determine the convergence or divergence of the series. $$ \frac{1}{201}+\frac{1}{208}+\frac{1}{227}+\frac{1}{264}+\cdots \cdot $$
Problem 81
In Exercises 81-84, give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.) A monotonically increasing sequence that converges to 10
Problem 81
Determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. $$ \sum_{n=1}^{\infty} \frac{10 n+3}{n 2^{n}} $$
