# Are the given infinite geometric series below has a sum? If they do, how to find it? (1) 100 + 110 + 121 + ..... (2) 100 + 10 + 1 + ..... (3) 0.3 + 0.6 + 1.2 + .....

Jul 5, 2018

Only sum (2) converges, and the value is $\frac{1000}{9}$

#### Explanation:

An infinite series can converge to a finite value if and only if the terms tend to zero. Otherwise, we are adding terms that get bigger and bigger (or, in the best hypothesis, terms that remain constant), and the sum will grow indefinitely.

So, since the terms in sums (1) and (3) grow bigger and bigger, both sums will diverge to $\setminus \infty$.

Sum (2), on the other hand, has terms that get smaller and smaller: we have

$100 + 10 + 1. . . = 100 \left(1 + \frac{1}{10} + \frac{1}{100.} . .\right) = 100 {\sum}_{n = 0}^{\infty} {\left(\frac{1}{10}\right)}^{n}$

and any goemetric series

${\sum}_{n = 0}^{\infty} {a}^{n}$

converges to $\frac{1}{1 - a}$ if $0 < a < 1$

So, in this case,

$100 {\sum}_{n = 0}^{\infty} {\left(\frac{1}{10}\right)}^{n} = 100 \left(\frac{1}{1 - \frac{1}{10}}\right) = 100 \cdot \frac{1}{\frac{9}{10}} = 100 \cdot \frac{10}{9} = \frac{1000}{9}$