Geometric Series Convergence

GS-Test

1. Given ${\sum }_{n=i}^{\infty }{a}_i{r}^n$, $a_i,r\in \mathbb{R},a_i\ne 0$ is a Geometric Series
2. If $|r|<1$ then, $\sum a_i{r}^n$ converges to $\frac{a}{1-r}$
3. If $|r|\ge 1$ then, $\sum a_i{r}^n$ diverges

GS Test Example

Prove ${\sum }_{n=4}^{\infty }((\frac{1}{2}{)}^n-7(\frac{6}{11}{)}^n)$ converges

• $={\sum }_{n=4}^{\infty }(\frac{1}{2}{)}^n+{\sum }_{n=4}^{\infty }-7(\frac{6}{11}{)}^n$
• Note that ${\sum }_{n=4}^{\infty }(\frac{1}{2}{)}^n$ is a Geometric Series with:
  • $a=(\frac{1}{2}{)}^4$
  • $r=(\frac{1}{2})$
• Then, the series converges to $\frac{a}{1-r}=\frac{7}{5}(\frac{6^4}{11^3})$ by Partial Sums for Geometric Series formulas
• Note that ${\sum }_{n=4}^{\infty }-7(\frac{6}{11}{)}^n$ is a Geometric Series with;
  • $a=-7(\frac{6}{11}{)}^4$
  • $r=(\frac{6}{11})$
• Then, the series converges to $\frac{a}{1-r}=(\frac{1}{2}{)}^4$ by Partial Sums for Geometric Series formulas
• Then, the entire series converges to $(\frac{1}{2}{)}^4-\frac{7}{5}(\frac{6^4}{11^3})$