Student Question

`sum_(n=1)^oo (-1)^(n+1)/(n+1)` Determine the convergence or divergence of the series.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`sum_(n=1)^oo ((-1)^(n+1))/(n+1)`

Take note that an alternating series

`sum` `a_n = sum (-1)^(n+1) b_n`

is convergent if the following conditions are satisfied.

(i)   `b_n` is decreasing, and

(ii)  ` lim_(n->oo) b_n=0` .

In the given alternating series, the bn is:

`b_n = 1/(n+1)`

Then, check if the values of bn decrease as n increases by 1.

`n=1` , `b_n = 1/2`

`n=2` , `b_n=1/3`

`n=3` , `b_n=1/4`

`n=4` , `b_n=1/5`

So bn is decreasing.

Also, take the limit of bn as n approaches infinity.

`lim_(n->oo) b_n = lim_(n->oo) 1/(n+1) = 0`

Since the result is zero, the second condition is satisfied too.

Therefore, by Alternating Series Test, the given series is convergent.

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial