## Test the convergence and absolute convergence of the alternating harmonic series.

1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...

## Test the convergence and absolute convergence of the series:

1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+...

## Test the convergence and absolute convergence of the series:

\sum_{n=2}^{\infty}(-1)^{n+1}\frac{1}{log\,n}

**Solution: (i) The given series**

\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n}

**is convergent by Leibnitz’s test.**

Now, \sum_{n=1}^{\infty}\left | u_n \right |=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...=\sum_{n=1}^{\infty}\frac{1}{n}

**is not convergent (As p=1)**

**Hence the given series is not absolutely convergent. This is an example of conditionally convergent series.**

**(ii) The given series**

\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n^2}

**is convergent by Leibnitz’s test.**

Also, \sum_{n=1}^{\infty}\left | u_n \right |=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...=\sum_{n=1}^{\infty}\frac{1}{n^2}

**is convergent (As p = 2 > 1) Hence the given series is absolutely convergent.**

**The given series**

\sum_{n=2}^{\infty}(-1)^{n+1}U_n

**Here u_n=\frac{1}{log\,n}**

**Now log x is an increasing function**

\forall\; x>0

:log(n+2)>log(n+1)

or \;\;\frac{1} {log(n+2)}<\frac{1} {log(n+1)}

\therefore u_{n+1} \leq u_n

**Also \lim_{n\rightarrow \infty}u_n=\lim_{n\rightarrow \infty}\frac{1}{log\,n}=0 **

**Hence by Leibnitz’s test , the given series is convergent.Now for absolute convergence , consider**

\sum_{n=1}^{\infty}\left | u_n \right | = \sum_{n=1}^{ \infty} \frac{1} {log\,n}

**It is a divergent series Hence the given series is not absolutely convergent. This is an exampleof conditionally convergent series.**

keywords

absolute convergence

determine whether the series is absolutely convergent, conditionally convergent, or divergent.

absolute convergence test

every absolutely convergent series is

absolutely convergent series examples

absolute convergence vs conditional convergence

**what is absolute convergence of the series**

A series Σ a_n converges absolutely if the series of the absolute values, Σ |a_n| converges

**what is conditionally convergent series**

A series that converges, but does not converge absolutely, converges conditionally.