Homework:

1.

Let $a_n = \frac{e^n-e^{-n}}{e^n+e^{-n}}$. Compute $\lim a_n$, and prove your answer using the definition of convergence.

2.

Prove, using the definition of divergence, that if $a_n = \frac{n^2-4}{n+5}$, then a_n diverges to $\infty$.

3.

Using the defintion of convergence, prove the following: Let a_n and b_n be sequences and define c_n = a_n+b_n. If $\lim a_n = L_1$ and $\lim b_n = L_2$, then $\lim c_n = L_1 + L_2$.

4.

Assume that -1 < r < 1, and consider the geometric series $\sum_{n=0}^{\infty} ar^n$. Write down the nth partial sum S_n. Prove that $S_n = \frac{a(1-r^{n+1})}{1-r}$. Prove that the series $\sum_{n=0}^{\infty} ar^n$ converges to $\frac{a}{1-r}$.

5.

Compute $\sum_{n=0}^{\infty} \frac{1}{e^n}$ (Hint, use what you just proved, or tried to prove, about the sum of a geometric series).

6.

Compute $\sum_{n=0}^{\infty} \frac{2^{k+3}}{e^{k-3}}$

7.

Determine whether each of the following series converges or diverges

8.

• $\sum_{n=1}^{\infty} \frac{(2n)!}{(n!)^3}$

• $\sum_{n=1}^{\infty} \frac{n}{\sqrt{n!}}$

• $\sum_{n=1}^{\infty} \frac{1+1/n}{3+1/n^2}$

• $\sum_{n=1}^{\infty} \frac{5+\sin(\sqrt{n})}{\sqrt{n}}$