0
$\begingroup$

Find the sum to $n$ terms as well as the sum to infinity of the series:

$$\frac{1}{3} + \frac{1}{4\cdot 2!} + \frac{1}{5\cdot 3!} +\cdots.$$

I was trying this question many times but I didn't get the answer. I was using geometric series formula, but it was not applicable here...I don't know where I have to start, and I don't know any hint about this series....

If anybody helps me, I would be very thankful to him.

$\endgroup$
  • 1
    $\begingroup$ Hint: this looks like the integral of $x^2e^x$ at $x=1$...or something close to that. $\endgroup$ – lulu Aug 17 '17 at 13:10
8
$\begingroup$

Each term is $\frac1{(n+2)n!}$.

Write this as $\frac{n+1}{(n+2)!} =\frac{(n+2)-1}{(n+2)!} =\frac1{(n+1)!}-\frac1{(n+2)!} $ and the sum telescopes.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.