Question

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

Answer 1

Let $S_n$ be the $n$th partial sum of a geometric sequence with common ratio $r$ and first term $a$. 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 $r$ gives

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

and subtracting $rS_n$ from $S_n$ 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 $0$, then $r^n\to0$ as $n\to\infty$ and so the sum approaches

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

Answer 2

Answer:

The formula for the sum is Sn= a1(1/r^n)/(1-r)

A fraction raised to a large power approaches zero.

The sum is a1 divided by the difference of 1 and r.