Calculate (express without infinite sum): $\frac{2}{1\cdot3}-\frac{4}{2 \cdot 9} + \frac{8}{3\cdot 27}- \frac{16}{4 \cdot 81} + ...$

Question

This is no homework, it's a task from an old exam and I'm wondering how it's solved correctly.

Calculate (express without infinite sum): $$\frac{2}{1\cdot3}-\frac{4}{2 \cdot 9} + \frac{8}{3\cdot 27}- \frac{16}{4 \cdot 81} + ...$$

We aren't allowed to use the sum symbol, so to be honest I got no idea how to do it.

I assume we are actually supposed to say what the limit of that is going to be? Because it's saying "calculate". I cannot imagine we can calculate this exactly, so I believe it's really the limit.

I have realized the signs are always switching, so for this we already got $(-1)^{n}$. Furthermore, the numerator will be greater than the denominator. This means this will always become smaller, will go towards zero.

But if we take the total sum / difference, we won't get zero because the beginning isn't small...

In total we will get to $0.5$ I believe.

But that way I described, would it count as solution and is it correct at all?

Answer 1

$S(x) = \displaystyle \sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}2^n}{n3^n}=-\displaystyle \sum_{n=1}^{\infty} \dfrac{x^n}{n}, x = -\dfrac{2}{3}$. You can differentiate $S$ and integrate it.Can you continue? or using Log.