Prove that: ∑n=1∞n2(n−1)2n=20

Autohelmvt

Autohelmvt

Answered

2022-01-24

Prove that:
n=1n2(n1)2n=20

Answer & Explanation

Troy Sutton

Troy Sutton

Expert

2022-01-25Added 13 answers

Note that ntn=11t for every |t|<1 hence, differentiaiting this twice and three times,
nn(n1)tn2=2(1t)3,
nn(n1)(n2)tn3=6(1t)4
For t=12, this reads
nn(n1)12n2=223
nn(n1)(n2)12n3=624
which implies
nn(n1)12n=14223=4
nn(n1)(n2)12n=18624=12
Finally,
n2(n1)=2n(n1)+n(n1)(n2)
hence
nn2(n1)12n=2nn(n1)12n+nn(n1)(n2)12n=24+12
This approach can be made shorter if one notices once and for all that, for every nonnegative k,
nn(n1)(nk)12n=2(k+1)!
egowaffle26ic

egowaffle26ic

Expert

2022-01-26Added 7 answers

For |x|<1 we have that n=0xn=11x hence differentiating both sides we get
n=0(n+1)xn=n=1nxn1=1(1x)2
while differentiating once more
n=0(n+2)(n+1)xn=2(1x)3
and once again
n=0(n+3)(n+2)(n+1)xn=6(1x)4
Next we express n2(n1) as a linear combination of (n+3)(n+2)(n+1),(n+2)(n+1),(n+1) and 1:
n2(n1)=(n+3)(n+2)(n+1)7(n+2)(n+1)+10(n+1)2
and hence
n=0n2(2n1)xn=6(1x)414(1x)3+10(1x)221x
and setting x=12 we obtain that
n=0n2(n1)2n=61648+1044=20

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?