Question

Can anyone help me pls? Does the infinite geometric series diverge or converge? Explain
1. 1/5+1/15+1/45+1/135

2. 7+28+112+448

Answer 1

if the ratio is less than one it converges, and the sum is easy to find

Answer 2

\[\frac{1}{5}+\frac{1}{15}+\frac{1}{45}+\frac{1}{135}+\cdots=\frac{1}{5}\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots\right)=\frac{1}{5}\sum_{n=0}^\infty \frac{1}{3^n}\]

Answer 3

@SithsAndGiggles so basically it's divergent?

Answer 4

Well the second one is, but the first converges.

Answer 5

The first series doesn't converge because it has a sum. The converse is true; that it has a sum because it converges.

The reason it converges is because of what satellite mentioned. A geometric series is of the form \(\sum ar^n\), where \(r\) is the common ratio between terms. If \(|r|<1\), the series converges. Otherwise the series diverges.

I showed that for the first series, the ratio is \(\dfrac{1}{3}\), which is less than 1, so ... convergent.

For the second series, whatever the ratio may be, there's no way it's smaller than 1. Each successive term is increasing. The series will not converge to a finite number.

Answer 6

@SithsAndGiggles thanks so much i got it! :)