How do I calculate this sum, using power series? $$\sum_{n\ge0}{{\frac{(n+1)(n+2)}{3^n}}}$$ I want just some hints to get me on the right track
Asked
Active
Viewed 53 times
2
-
Looks the the second derivative of a power series in x evaluated at 1. – Charlie Frohman Nov 24 '16 at 18:31
-
http://math.stackexchange.com/questions/593996/how-to-prove-sum-n-0-infty-fracn22n-6 – lab bhattacharjee Nov 24 '16 at 18:31
1 Answers
3
Hint:
$$(n+2)(n+1)x^n$$
is the second derivative of $x^{n+2}$. Therefore,
$$\sum_{n \ge 0} (n+2)(n+1)x^n \tag{1}$$
is the second derivative of $$ \sum_{n \ge 0} x^{n+2} = \frac{x^2}{1-x}. \tag{2} $$
See if you can use (2) to get a closed form for (1). Then plug in a specific value of $x$.
Caleb Stanford
- 45,895