Alternating series:

A series of the form

\(\sum_{n=1}^\infty(-1)^{n+1}na_n=a_0-a_1+a_2-a_3+...\)

where either all \(a_n\) are positive or all \(a_n\) are negative, is called an alternating series.

The alternating series test then says: if \(|a_n|\) decreases monotonically and \(\lim_{n\rightarrow\infty}a_n=0\) then the alternating series converges.

Moreover, let L denote the sum of the series, then the partial sum

\(S_k=\sum_{n=0}^k(-1)^na_n\)

approximates L with error bounded by the next omitted term:

\(|S_k-L|\leq|S_k-S_{k+1}|=a_{k+1}\)

i) \(a_n=ne^{-n}\)

\(=\frac{n}{e^n}>0\)

ii) \(n\to\infty\frac{n}{e^n}\)

\(n\to\infty\frac{1}{e^n}\)

\(=\frac{1}{\infty}\)

\(=0\)

iii) \(a_{n+1}

Hence the alternating series will converge.

A series of the form

\(\sum_{n=1}^\infty(-1)^{n+1}na_n=a_0-a_1+a_2-a_3+...\)

where either all \(a_n\) are positive or all \(a_n\) are negative, is called an alternating series.

The alternating series test then says: if \(|a_n|\) decreases monotonically and \(\lim_{n\rightarrow\infty}a_n=0\) then the alternating series converges.

Moreover, let L denote the sum of the series, then the partial sum

\(S_k=\sum_{n=0}^k(-1)^na_n\)

approximates L with error bounded by the next omitted term:

\(|S_k-L|\leq|S_k-S_{k+1}|=a_{k+1}\)

i) \(a_n=ne^{-n}\)

\(=\frac{n}{e^n}>0\)

ii) \(n\to\infty\frac{n}{e^n}\)

\(n\to\infty\frac{1}{e^n}\)

\(=\frac{1}{\infty}\)

\(=0\)

iii) \(a_{n+1}

Hence the alternating series will converge.