How do you find the sum of the infinite geometric series 1+1/5+1/25+....?

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Answer 1 · 2015-11-09T08:02:55

Divide terms to find $r$ and then apply the geometric series formula to find the sum to be $5/4$

A geometric series is a series of the form
$a+ar+ar^2+ar^3+...$
where $a$ is an initial value and $r$ is a common ratio between terms.

In general, the sum of a geometric series is
$a + ar + ar^2 + ... + ar^(n-1) = a(1-r^n)/(1-r)$ (see below for derivation)

If $|r| < 1$ then as $n ->oo, r^n->0$ so the formula reduces to

$a + ar + ar^2 + ... = a/(1-r)$

Now, in order to find $r$ , it suffices to divide any term in the series after the first by the term prior to it, as
$(ar^n)/(ar^(n-1))=r$

So, picking the second and first terms, for this series we get
$r = (1/5)/1 = 1/5$

As the first term in the series gives us $a=1$ we get the final sum
$1/(1-(1/5)) = 1/(4/5) = 5/4$

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What follows is a short derivation for the geometric series formula, and is not required for understanding the above solution.

Let $S_n$ be the $n$ th partial sum of a geometric series with ratio $r$ and first term $a$ , that is,
$S_n = a + ar + ar^2 + ... + ar^(n-1)$

$=>rS_n = ar + ar^2 + ... + ar^n$

$=> S_n - rS_n = a - ar^n$

$=> S_n(1-r) = a(1-r^n)$

$=> S_n = a(1-r^n)/(1-r)$