Question

The sum of the infinite geometric series is finite if and only if:

• A. $|r|<1$
• B. $|r|\ge 1$
• C. $|r|\le 1$
• D. $|r|=1$

Answer 1

The correct option is A $|r|<1$

Solution:

Let $a$ be first term and $r$ be common ratio

${\sum }_{i=1}^n{a}_i=a\times \frac{1-r^n}{1-r}$

Now, if $n\to \infty$

${\sum }_{i=1}^{\infty }{a}_i=\frac{a}{1-r}$ where, $|r|<1$ i.e. $-1<r<1$

Hence, A is the correct option.