To get points for this, please complete all parts to the problem, and show ALL w

Question

To get points for this, please complete all parts to the problem, and show ALL work. Bonus points: Write it down on paper and take a picture of it showing all of your steps. Please prove your work completely. If it is completed entirely with good explinations, and work, you will get the points. I really need to understand this so it would be very helpful if you show all steps. I'm not expecting it to be answered in 4 hours. If you're working on it, please post here, and I'll gladly wait longer. If you use an oustside source, please provide it.

http://i.imgur.com/IXZzxSF.jpg?1

Explanation / Answer

(a) sum (over n = 0 to +inf) (n^k / k^n) = 1 + 1/k + 2^k/k^2 + 3^k/k^3 + .... = call it s(k) ...  

      The ratio of the (n+1)th term to the nth term is = ( (n+1)^k / k^(n+1) ) / ( n^k / k^n ) = {(n+1)/n}^k / k = [ 1 + 1/n ]^k / k  

      The value of this ratio as n -> +inf is lim (n -> +inf) [ (1 + 1/n)^k / k ] = lim (n -> +inf) [ (1 + 1/n)^k / k ] =   

       lim (n -> +inf) [ { (1 + 1/n)^(kn/n) } / k ] = lim (n -> +inf) [ { { (1 + 1/n)^n }^(k/n)) } / k ] = lim (n -> +inf) [ { { e }^(k/n)) } / k ].  

       As n -> +inf, e^(k/n) -> e^0 = 1 for finite k. So, the ratio -> 1/k. Now, 1/k < 1 if k > 1. Hence, the series converges for k > 1

Rest coming soon....

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