Chapter 7: Problem 112
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The series \(\sum_{n=1}^{\infty} \frac{n}{1000(n+1)}\) diverges.
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Writing In Exercises 89 and 90 , use a graphing utility to determine the first term that is less than 0.0001 in each of the convergent series. Note that the answers are very different. Explain how this will affect the rate at which each series converges. $$ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} $$
Let \(a_{n}=\frac{n+1}{n}\). Discuss the convergence of \(\left\\{a_{n}\right\\}\) and \(\sum_{n=1}^{\infty} a_{n}\).
(a) Show that \(\int_{1}^{n} \ln x d x<\ln (n !)\) for \(n \geq 2\).
(b) Draw a graph similar to the one above that shows
\(\ln (n !)<\int_{1}^{n+1} \ln x d x\)
(c) Use the results of parts (a) and (b) to show that
\(\frac{n^{n}}{e^{n-1}}
Determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty}(1.075)^{n} $$
Find the values of \(x\) for which the infinite series \(1+2 x+x^{2}+2 x^{3}+x^{4}+2 x^{5}+x^{6}+\cdots\) converges. What is the sum when the series converges?
What do you think about this solution?
We value your feedback to improve our textbook solutions.
