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:

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

\(a_n=\frac{1}{n+1}\)

\(a_{n+1}=\frac{1}{n+2}\)

Therefore \(a_{n+1}

So \(a_n\) is monotonically decrea \(\sin g\)

\(a_n>0\) for all \(n=1,2,3,...\)

\(n\to\infty\)

\(=0\)

So the series converges.

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:

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

\(a_n=\frac{1}{n+1}\)

\(a_{n+1}=\frac{1}{n+2}\)

Therefore \(a_{n+1}

So \(a_n\) is monotonically decrea \(\sin g\)

\(a_n>0\) for all \(n=1,2,3,...\)

\(n\to\infty\)

\(=0\)

So the series converges.