Chapter 7: Problem 11
Verify that the infinite series diverges. $$ \sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1} $$
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
Use the formula for the \(n\) th partial sum of a geometric series $$\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}$$ You go to work at a company that pays \(\$ 0.01\) for the first day, \(\$ 0.02\) for the second day, \(\$ 0.04\) for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working (a) 29 days, (b) 30 days, and (c) 31 days?
Inflation If the rate of inflation is \(4 \frac{1}{2} \%\) per year and the average price of a car is currently \(\$ 16,000,\) the average price after \(n\) years is \(P_{n}=\$ 16,000(1.045)^{n}\) Compute the average prices for the next 5 years.
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \ln \frac{1}{n} $$
Let \(\left\\{x_{n}\right\\}, n \geq 0,\) be a sequence of nonzero real numbers such that \(x_{n}^{2}-x_{n-1} x_{n+1}=1\) for \(n=1,2,3, \ldots .\) Prove that there exists a real number \(a\) such that \(x_{n+1}=a x_{n}-x_{n-1},\) for all \(n \geq 1 .\)
(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}}
What do you think about this solution?
We value your feedback to improve our textbook solutions.
