SOLUTION: for each series, determine if the series converges or diverges. list at least the first 5 partial sums for each series. if the series converges, state the value. show graphical sup

Algebra ->  Sequences-and-series -> SOLUTION: for each series, determine if the series converges or diverges. list at least the first 5 partial sums for each series. if the series converges, state the value. show graphical sup      Log On


   

Question 417410: for each series, determine if the series converges or diverges. list at least the first 5 partial sums for each series. if the series converges, state the value. show graphical support for you conclusions.
infinite above sigma, k=1 under sigma, (1/(k-1!) on the side
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
I think you're implying the sum sum%28%281%2F%28k-1%29%21%29%2C+k=1%2C+infinity%29.

We can use the ratio test. We wish to show that

lim%28k-%3Einfinity%2C+abs%28%281%2Fk%21%29%2F%281%2F%28k-1%29%21%29%29%29 exists and is less than 1. This limit is equal to



Hence, the series converges. If you know the Taylor series e%5Ex+=+sum%28%28x%5Ek%29%2Fk%21%2C+k=0%2Cinfinity%29 and that the sums sum%28%281%2F%28k-1%29%21%29%2C+k=1%2C+infinity%29 and e%5Ex+=+sum%28%28x%5Ek%29%2Fk%21%2C+k=0%2Cinfinity%29 are equivalent when x=1, we conclude that the series converges to e.