What is the sum of the infinite geometric series 1 + $\frac{1}{5}$ $1/5$ + $\frac{1}{25}$ $1/25$ +... ?

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What is the sum of the infinite geometric series 1 + $\frac{1}{5}$ $1/5$ + $\frac{1}{25}$ $1/25$ +... ?

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Answer 1 · 2015-05-23T21:59:49

$1 + \frac{1}{5} + \frac{1}{25} + \dots = \frac{5}{4}$ $1+1/5+1/25+cdots=5/4$

For the infinite geometric series $S = a + a q + a q^{2} + a q^{3} + \dots$ $S=a+aq+aq^2+aq^3+cdots$ we can use the formula $S = \frac{a}{1 - q}$ $S=a/(1-q)$ as long as $| q | < 1$ $|q|<1$ .
In our case $a = 1, q = \frac{1}{5}$ $a=1, q=1/5$ and $| q | = ∣ ∣ ∣ \frac{1}{5} ∣ ∣ ∣ = \frac{1}{5} < 1$ $|q|=|1/5|=1/5<1$ so
$S = \frac{1}{1 - \frac{1}{5}} = \frac{1}{\frac{4}{5}} = \frac{5}{4}$ $S=1/(1-1/5)=1/(4/5)=5/4$ .

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