# How do I find the sum of the infinite geometric series 1/2 + 1 + 2 + 4 +... ?

Jun 5, 2018

see below

#### Explanation:

for a GP the sum to infinity

${S}_{\infty} = \frac{a}{1 - r}$

$a = \text{ first term, "r=" the common ratio }$

only exists if

$| r | < 1$

for the given GP

$\frac{1}{2} + 1 + 2 + 4 + . .$

$r = \frac{1}{\frac{1}{2}} = 2$

$\therefore | r | > 1$

$\implies \text{ sum to infinity does not exist}$

Jun 9, 2018

${S}_{n} = \frac{1}{2} {\Sigma}_{i = 0}^{n} \left({2}^{i}\right) \to \infty$ as $n \to \infty$, i.e. the sum is indefinite.

#### Explanation:

You can write the sum of n first terms in the geometric series as ${S}_{n} = \frac{1}{2} {\Sigma}_{i = 0}^{n} \left({2}^{i}\right)$
It's a well known fact that ${2}^{n} \to \infty$ as $n \to \infty$. Therefore, as the individual terms go towards infinity, so must the sum.

Therefore ${S}_{n} = \frac{1}{2} {\Sigma}_{i = 0}^{n} \left({2}^{i}\right) \to \infty$ as $n \to \infty$