The only you have to do is writing the series ‘term by term’…
… and the value of the sum is quite evident…
Kind regards
Dear Molly…
… I’m afraid have made a great mistake!… The series you have proposed has termes with alternating sign and decreasing in absolute value. But necessary condition for convergence is that the general term tend to 0 and in this case it isn’t. So the series doesn’t converge… and your thanks to me are totally not deserved …
I apologize and promit never to repeat such a terrible mistake!…
Very sorry! …
Effectively the general term of the series is…
… so that is…
So the series converges and its sum is what you have computed: . Tonight i discovered to have [probably] done a bad suggestion to Molly and for that i was 'furious' ... This story contains an useful teaching: hurry gives bad suggestions...
Kind regards