Consider the sequence \displaystyle{ \lbrace a_n \rbrace = \left\lbrace \frac{2}{n^2+2n} \right\rbrace } .
\infty or -\infty ; otherwise, enter DNE if the limit does not exist.

- The limit of this sequence is
\displaystyle{ \lim_{n \to \infty} a_n = } . - 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) = } .

Consider the sequence \displaystyle{ \lbrace b_n \rbrace = \left\lbrace \ln\left( \frac{n+1}{n} \right) \right\rbrace } .
\infty or -\infty ; otherwise, enter DNE if the limit does not exist.

- The limit of this sequence is
\displaystyle{ \lim_{n \to \infty} b_n = } . - 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) = } .

Suppose \lbrace c_n \rbrace is a sequence.

- If
\displaystyle{ \lim_{n \to \infty} c_n = 0 } , then the series\displaystyle{ \sum_{n=1}^{\infty} c_n } choose must may or may not cannot - If
\displaystyle{ \lim_{n \to \infty} c_n \ne 0 } , then the series\displaystyle{ \sum_{n=1}^{\infty} c_n } choose must may or may not cannot - If the series
\displaystyle{ \sum_{n=1}^{\infty} c_n } converges, then\displaystyle{ \lim_{n \to \infty} c_n } choose must may or may not cannot 0 .

You can earn partial credit on this problem.