simplifying an equation with fractions

Question

I am trying to understand a proposed solution posted here (by user17762) to a problem in Feller's book Introduction to probability and its applications, and there is a step that I do not understand. Can anyone please explain what happens here:

How does the right side follow from the left side?

Thank you so much in advance.

Answer 1

This is a particular case of the geometric series: $$|r|<1\implies1+r+r^2+r^3+\cdots=\frac1{1-r}.$$

Answer 2

Hint:

If $|x|<1$ then :

$$1+x+x^2+... = {1\over 1-x}$$

Answer 3

It is a geometric sum; you can prove that

$\sum_{k=0}^\infty a^k =\frac{1}{1-a}$ for $|a|<1$

Answer 4

Working with a geometric series like this, we can attain either a sum to a finite number of terms, or a sum to infinity. In general: $$S_\infty=\frac{a}{1-r}$$ where $a$ is the first term and $r$ is the multiplier. Here $a=1$ and $r=\frac14$, and so $S_\infty=\frac{1}{1-\frac14}(=\frac43)$