Homework # 11 - Math 1150
The problems on this homework must be done alone. The honor code is in effect.
You may submit the solutions twice, if you wish.
First Name: Last Name:
1. Find the limit of the series \(\sum_{n = 1}^{\infty} \frac{9}{2}(\frac{1}{10})^n\)
a. \(\frac{9}{10}\)
b. \(4.75\)
c. \(5\)
d. \(\frac{11}{2}\)
e. This series diverges.
Note: The \(n\)th term test is another name for the divergence criterion.
2. Consider the series \(\sum_{n = 2}^{\infty} \frac{1}{n\ln n}\)
a. The series converges by the \(n\)th term test.
b. The series diverges by the \(n\)th term test.
c. The series converges by the integral test.
d. The series diverges by the integral test.
3. Consider the series \(\sum_{n = 1}^{\infty} \frac{3^n}{4^n - 1}\)
c. The series converges by the LCT.
d. The series diverges by the LCT.
4. Consider the series \(\sum_{n = 1}^{\infty} (-1)^{n+1} \frac{n}{n+1}\)
d. The series converges by the LCT.
5. Consider the series \(\sum_{n = 1}^{\infty} \frac{n^2 + n + 3}{2n^2 + 10 n}\)
In questions 6 through 9, determine whether the series converge or diverge. Practice writing out your work carefully for each problem (though you don't have to submit it). State which test you are using, show all your computations, and state your final answer (converges/diverges).
6. \(\sum_{n = 1}^{\infty} n (\frac{2}{3})^n\)
a. The series converges.
b. The series diverges.
7. \(\sum_{n = 1}^{\infty} \frac{1}{n+\sqrt{n}\)
8. \(\sum_{n = 1}^{\infty} \frac{(n!)^2}{(3n)!}\)
9. \(\sum_{n = 1}^{\infty} \frac{1}{3n}\)
10. Can the series \(\sum_{n = 1}^{\infty} (-1)^n\frac{1}{n^2}\) be rearranged to get any limit you want?
a. Yes, because it converges absolutely.
b. No, because it converges absolutely.
a. Yes, because it converges conditionally.
b. No, because it converges conditionally.
11. Can the series \(\sum_{n = 1}^{\infty} (-1)^n\frac{1}{\sqrt{n}}\) be rearranged to get any limit you want?