I do not know how to find the value of this sum:
n∑k=1k2k+n2n
(Yes, the last term is added twice).
Of course I've already plugged it to wolfram online, and the answer is 2−12n−1
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)=k∑n=0(cx)n=1−(cx)k+11−cx
Taking the derivative of both sides:
f′n(x)=ck−1∑n=0n(cx)n=c(k+1)(cx)kcx−1−c((cx)k+1−1)(cx−1)2
For your problem, just plug in c=2−1 and x=1, and then add the final term.
Or you could just use the following identity:
n∑i=1if(i)=n∑j=1(n∑i=jf(i))
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