$$\frac{7}{8} – \frac{49}{64} + \frac{343}{512} – \frac{2401}{4096} +\cdots$$
So I am not sure how to replicate this into summation notation but I have several ways of doing it and I have no idea which one is the right one because clearly some (all) don't work and only specific versions of the replication do work.
$$\sum_{n=0}^\infty \frac{\sqrt7}{\sqrt8}^{(2(n+1))}$$
$$\sum_{n=0}^\infty -1^n \cdot \frac{7}{8}^{n+1}$$
$$\sum_{n=1}^\infty -1^{n+1} \cdot \frac{7}{8}^{n}$$
And several others, I would jsut be wasting my time because I am sure I am doing these all so horribly wrong as to embarrass myself.
Anyways trying to use the form $\dfrac{1}{1-r}$ doesn't give the correct answer on any of these.
Best Answer
$$\frac{7}{8} - \frac{49}{64} + \frac{343}{512} - \frac{2401}{4096} +\cdots=\frac{7}{8}\sum_{n=0}^{\infty}\left(-\frac{7}{8}\right)^n=\frac{7}{8}\frac{1}{1-(-7/8)}$$