Problem 37 on the Practice Test - I need help.

Forum for the GRE subject test in mathematics.
Eschewy
Posts: 4
Joined: Thu Mar 20, 2008 11:57 pm

Problem 37 on the Practice Test - I need help.

Postby Eschewy » Fri Mar 21, 2008 12:12 am

I need some quick help on problem 37 from the practice test:

What is the sum of k^2/k! as k = 0 -> infinity?

The booklet claims the answer is 2*e, and I know for a fact that the sum of 1/k! is e, but I can't logically get from one to the other. I've tried looking at Taylor series, but it's no use. Any help would be greatly appreciated.

Eschewy
Posts: 4
Joined: Thu Mar 20, 2008 11:57 pm

The Answer

Postby Eschewy » Fri Mar 21, 2008 2:29 am

False alarm, guys. I found the answer on another message board.

First note that, for k >= 1, k^2/k! = k/(k-1)!. By a change of
variable we may write sum(k=1 to oo) k/(k-1)! = sum(k=0 to oo) (k+1)/
k! = sum(k=0 to oo) 1/k! + sum(k=0 to oo) k/k!. The first term is e.
For the second term note that for the case k=0 the summand is zero, so
we can write this as sum(k=1 to oo) k/k! = sum(k=1 to oo) 1/(k-1)! =
sum(k=0 to oo) 1/k!, where the last equality follows from another
change of variable. So this term is also e.




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