Find #sum_(n=1)^oo n^2/3^n = # ?

Question

220 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-06T21:38:42

See below.

We will solve the last item.

We have

#x d/(dx)(x d/(dx)(sum_(n=1)^oo x^n)) = sum_(n=1)^oo n^2 x^n#

now considering that for #abs x < 1#

#sum_(n=1)^oo x^n = 1/(1-x)-1# we have:

#x d/(dx)(x d/(dx)( 1/(1-x)-1)) = (x(x+1))/(1-x)^3#

then for #x = 1/3# we have

#sum_(n=1)^oo n^2/3^n = 1/3(1+1/3)/(2/3)^3 = 3/2#

The first item is a consequence and is left as an exercise.