§8.3 Geometric sums

Introduction

Finite Geometric Series

Infinite Geometric Series

For the geometric series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$ with ratio $\frac{1}{2}$, the partial sum $S_4$ equals

For the geometric series with first term $a$ and ratio $r$, the finite partial sum formula $S_n = \frac{a(1-r^n)}{1-r}$ follows most directly from

A series has first term $\frac{7}{5}$ and ratio $\frac{2}{5}$, so it reads $\frac{7}{5}+\frac{14}{25}+\frac{28}{125}+\cdots$. Its partial sums satisfy

As $n$ grows, the partial sums of $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$ approach the limit

For the series $2+\frac{8}{3}+\frac{32}{9}+\cdots$ with ratio $\frac{4}{3}$, the partial sums

Compute the sum $1+\frac{1}{2}+\frac{1}{4}+\cdots+\left(\frac{1}{2}\right)^9$ of ten terms, rounded to three decimals.

Compute $2-\frac{8}{3}+\frac{32}{9}-\frac{128}{27}+\cdots+2\left(-\frac{4}{3}\right)^7$, the eight-term sum with $a = 2$ and $r = -\frac{4}{3}$, rounded to three decimals.

Compute the nine-term sum $\frac{1}{3}+1+3+9+\cdots+6561$, which has $a = \frac{1}{3}$ and $r = 3$, rounded to three decimals.

Compute $3+9+27+\cdots+3^5$, the five-term sum with $a = 3$ and $r = 3$, reporting the exact value as a decimal.

Compute the infinite sum $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$ with $r = -\frac{1}{2}$, rounded to three decimals.

Compute the infinite sum $\sum_{k=1}^{\infty}-2\left(-\frac{2}{3}\right)^k$, which starts $\frac{4}{3}-\frac{8}{9}+\cdots$, rounded to three decimals.

If $S = 1+\frac{1}{2}+\frac{1}{4}+\cdots = 2$, then the scaled series $2+1+\frac{1}{2}+\cdots$ equals $2S = 4$, and similarly $S' = 2+\frac{2}{3}+\frac{2}{9}+\cdots = 2\cdot\frac{1}{1-\frac{1}{3}} = 3$.