Consider the sequence $\displaystyle{ \lbrace a_n \rbrace = \left\lbrace \frac{2}{n^2+2n} \right\rbrace }$.
1. The limit of this sequence is $\displaystyle{ \lim_{n \to \infty} a_n = }$ .
2. The sum of all terms in this sequence is defined as the the limit of the partial sums, which means
$\displaystyle{ \sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} \bigg( }$ $\displaystyle{\bigg) = }$ .
Enter infinity or -infinity if the limit diverges to $\infty$ or $-\infty$; otherwise, enter DNE if the limit does not exist.
Consider the sequence $\displaystyle{ \lbrace b_n \rbrace = \left\lbrace \ln\left( \frac{n+1}{n} \right) \right\rbrace }$.
1. The limit of this sequence is $\displaystyle{ \lim_{n \to \infty} b_n = }$ .
2. The sum of all terms in this sequence is defined as the the limit of the partial sums, which means
$\displaystyle{ \sum_{n=1}^{\infty} b_n = \lim_{n \to \infty} \bigg( }$ $\displaystyle{\bigg) = }$ .
Enter infinity or -infinity if the limit diverges to $\infty$ or $-\infty$; otherwise, enter DNE if the limit does not exist.
Suppose $\lbrace c_n \rbrace$ is a sequence.
1. If $\displaystyle{ \lim_{n \to \infty} c_n = 0 }$, then the series $\displaystyle{ \sum_{n=1}^{\infty} c_n }$ converge. Hint: look back at parts 1 and 2.
2. If $\displaystyle{ \lim_{n \to \infty} c_n \ne 0 }$, then the series $\displaystyle{ \sum_{n=1}^{\infty} c_n }$ converge.
3. If the series $\displaystyle{ \sum_{n=1}^{\infty} c_n }$ converges, then $\displaystyle{ \lim_{n \to \infty} c_n }$ be equal to $0$.

You can earn partial credit on this problem.