Question

HELP ME! I have just started infinite geometric series and it's kicking my butt! I have no idea what to do. Attachment below.

Answer 1

@whpalmer4

Answer 2

For an infinite geometric series where \(|r| < 1\),

\[\sum_{i=1}^\infty a_i = \frac{a}{1-r}\]

Answer 3

If you remember the formula for a finite geometric series:

\[S_n = \sum_{i=1}^{n} a_i = \frac{a(1-r^n)}{1-r}\]You can figure out the infinite version by noticing that as \(n\rightarrow\infty\), \(r^n\rightarrow0\) (because \(|r| < 1\)) leaving you with just \[\frac{a(1-0)}{1-r} = \frac{1}{1-r}\]

But the kicker is that for that to converge, the absolute value of the common ratio must be less than 1, or the number just keeps growing.