Sum to infinity | Mathematico

Consider a geometric sequence with first term $a$ and common ratio $r$ .

If $| r | < 1$ , then the term $r^{n}$ gets very small as $n$ gets very large. This means that

\begin{aligned} lim_{n \to \infty} S_{n} & = lim_{n \to \infty} a \frac{1 - r^{n}}{1 - r} \\ = a \frac{1 - 0}{1 - r} \end{aligned}

Recall that $n$ is the number of terms in the sum, and so if we keep adding up the terms forever, we get

S_{\infty} = \frac{a}{1 - r}

This is the formula for the sum to infinity of a geometric sequence.

Remember: this formula only works when $- 1 < r < 1$ . Otherwise, the term $r^{n}$ only gets bigger as $n$ gets bigger, and the numerator in the $S_{n}$ formula doesn't become $1 - 0$ .

Use the formula for $S_{\infty}$ to find the sum of the sequences:

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$
$3 - 1 + \frac{1}{3} - \frac{1}{9} + \dots$