given
$$\sum_{n=0}^\infty x^{n} (n^{2} + n)$$
so using ratio test I have proven that it converges if and only if
$$|x| < 1$$
but I'm not sure how to evaluate this infinite sum.
so I thought about deriving it, but I'm not sure whether I should do it with respect to
x or n.
Best Answer
HINT:
$$\dfrac{d(x^n)}{dx}=nx^{n-1}$$
$$\dfrac{d^2(x^n)}{dx^2}=n(n-1)x^{n-2}$$
$$Ax\dfrac{d(x^n)}{dx}+Bx^2\dfrac{d^2(x^n)}{dx^2}=x^n[An+Bn(n-1)]=x^n[Bn^2+n(A-B)]$$
We need $A-B=1,B=1$
Now $\displaystyle\sum_{r=0}^\infty x^r=\dfrac1{1-x}$ for $|x|<1$