Mathematics: Explain why a geometric series with a ratio between zero and one converges and how you find the sum.

Augustine of Hippo. I am sure...

Explain why a geometric series with a ratio between

elizabethajih99

04.01.2022

Let

be the

th partial sum of a geometric sequence with common ratio

and first term

. So the sequence is $\{a,ar,ar^2,\cdots\}$ , and S_n

is

$S_n=a+ar+ar^2+\cdots+ar^{n-2}+ar^{n-1}$

Multiplying both sides by

gives

$rS_n=ar+ar^2+ar^3+\cdots+ar^{n-1}+ar^n$

and subtracting rS_n

from

gives

$S_n-rS_n=a+(ar-ar)+(ar^2-ar^2)+\cdots+(ar^{n-1}-ar^{n-1})-ar^n$
(1-r)S_n=a(1-r^n)

$S_n=a\dfrac{1-r^n}{1-r}$

If

, then $r^n\to0$ as $n\to\infty$ and so the sum approaches

$\displaystyle\lim_{n\to\infty}S_n=\dfrac a{1-r}$

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