I am struggling a bit with power series at the moment, and I don't quite understand what this question is asking me to do? Am I meant to form a power series from these, or simply evaluate that series? Any explanation/working is appreciated.
Using power series representation, calculate
$$\sum_{n=1}^\infty \frac{n2^n}{3^n}.$$
Best Answer
Recall that, in general, $$1 + x + x^2 + \cdots = \frac{1}{1 - x}, \quad |x| < 1.$$
Moreover, power series can be differentiated term by term. So, differentiating both sides of the equation above we get $$1 + 2x + 3x^2 + \cdots = \frac{1}{(1 - x)^2}, \quad |x| < 1.$$
Now, multiplying both sides by $x$ leads to $$x + 2x^2 + 3x^3 + \cdots = \sum_{n = 1}^\infty nx^n = \frac{x}{(1 - x)^2}, \quad |x| < 1.$$
However, in this case $x = 2/3 < 1$, so simply substitute $x = 2/3$ in formula above.