Tuesday, November 15, 2016

summation - How to compute this finite sum sumnk=1frack2k+fracn2n?




I do not know how to find the value of this sum:



nk=1k2k+n2n



(Yes, the last term is added twice).



Of course I've already plugged it to wolfram online, and the answer is 212n1



But I do not know how to arrive at this answer.




I am not interested in proving the formula inductively :)


Answer



Take the following:
fk(x)=kn=0(cx)n=1(cx)k+11cx
Taking the derivative of both sides:
fn(x)=ck1n=0n(cx)n=c(k+1)(cx)kcx1c((cx)k+11)(cx1)2

For your problem, just plug in c=21 and x=1, and then add the final term.



Or you could just use the following identity:
ni=1if(i)=nj=1(ni=jf(i))


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