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watchmath
 5 years ago
For this one I don't know the answer yet for sure
Find
\[\sum_{n=0}\infty \frac{1}{n!(n^4+n^2+1)}\]
I will be your fan if you can answer this :D.
watchmath
 5 years ago
For this one I don't know the answer yet for sure Find \[\sum_{n=0}\infty \frac{1}{n!(n^4+n^2+1)}\] I will be your fan if you can answer this :D.

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watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0I think I got some idea :D

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0I think the answer is \(\frac{3}{2}\).

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Here's an idea. Suppose that we want to find \[ \sum_{n=0}^\infty \frac{1}{n!(n+a)} \] for any number \(a\) not an integer \(\leq 0\), even a complex one. Start with the infinite series \[ e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, \] multiply by \(x^{a1}\), \[ x^{a1}e^x = \sum_{n=0}^\infty \frac{x^{n+a1}}{n!} \] and integrate \[ \int x^{a1}e^x = \int \sum_{n=0}^\infty \frac{x^{n+a1}}{n!} = \sum_{n=0}^\infty \frac{x^{n+a}}{n!(n+a)} \] Find the integral on the left, then you can plug in 1 for x and get the sum (watch your choice of a constant though, should always be zero I think). In a similar manner you could find \[ \sum_{n=0}^\infty \frac{1}{n!(n+a)(n+b)} \] etc. I'm sure there is a better way to do the problem you posted, but this occurred to me as a possible appoach. I didn't do the computation to get \(n^4+n^2+1\), but you could factor it over the complex numbers and go for it.

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0How can you go from \(\int x^{a1}e^x = \int \sum_{n=0}^\infty \frac{x^{n+a1}}{n!} = \sum_{n=0}^\infty \frac{x^{n+a}}{n!(n+a)}\) to \(\sum_{n=0}^\infty \frac{1}{n!(n+a)(n+b)}\)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0do you think this is something nice?

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0It turns out to be nice satellite. It can make your afternoon beautiful :D.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i mean say \[\frac{e}{2}\] or \[\frac{\pi^2}{6}\]

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.0I made a mistake. You are right satellite. It is \(e/2\).

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0It's \(\frac{e}{2}\), not \(\frac{3}{2}\)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0really? that was a total guess. really.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i just wrote it to say something.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0i am going to go play the lottery! what are the chances of that? pick a number. ok \[\frac{e}{2}\]!

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0commutant how did you get it?
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