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(1-r^n)/(1-r)`

If
`0<r<1`, then
`r^n\to0` as
`n\to\infty` and so the sum approaches


`\displaystyle\lim_(n\to\infty)S_n=\frac a{1-r}`

Answer 2

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.