4.5 Finite and Infinite Series

A geometric series is the sum of the terms of a geometric sequence. This section covers the sum of a finite geometric series, the conditions for an infinite geometric series to converge, and real-world applications.

---

4.5.1 Sum of a Finite Geometric Series

For a geometric sequence with first term $a_1$, common ratio $r$, and $n$ terms, the sum is:

$S_n=\frac{a_1(1-r^n)}{1-r},r\ne 1$

An equivalent form (useful when $r>1$) is:

$S_n=\frac{a_1(r^n-1)}{r-1},r\ne 1$

When $r=1$, all terms are equal to $a_1$, so $S_n=n\cdot a_1$.

Useful Variant

If the last term $a_n$ is known, the sum can be written as:

$S_n=\frac{a_1-r{a}_n}{1-r}$

This avoids computing $r^n$ when $a_n$ is already given.

Example 1

Find the sum of the first 6 terms of the geometric series $2+6+18+\dots$

Solution: $a_1=2$, $r=3$, $n=6$

$S_6=\frac{2(3^6-1)}{3-1}=\frac{2(729-1)}{2}=728$

---

4.5.2 Infinite Geometric Series

An infinite geometric series is $a_1+a_{1}r+a_1{r}^2+\dots$

Convergence Condition

The series converges to a finite sum if and only if $|r|<1$.

If $|r|\ge 1$, the terms do not approach zero and the series diverges (no finite sum).

Formula for Infinite Sum

$S_{\infty }=\frac{a_1}{1-r},|r|<1$

Example 2

Find $S_{\infty }$ for the series $12+4+\frac{4}{3}+\dots$

Solution: $a_1=12$, $r=\frac{1}{3}$, $|r|<1$ ✓

$S_{\infty }=\frac{12}{1-\frac{1}{3}}=\frac{12}{\frac{2}{3}}=18$

---

4.5.3 Repeating Decimals as Infinite Geometric Series

Every repeating decimal can be expressed as an infinite geometric series and converted to a fraction.

Example 3

Convert $0.\overline{15}=0.151515\dots$ to a fraction.

Solution: $0.151515\cdots =0.15+0.0015+0.000015+\dots$

Here $a_1=0.15$ and $r=0.01$, so $|r|<1$ ✓

$S_{\infty }=\frac{0.15}{1-0.01}=\frac{0.15}{0.99}=\frac{15}{99}=\frac{5}{33}$

Example 4

Convert $0.\overline{4}=0.444\dots$ to a fraction.

$0.444\cdots =0.4+0.04+0.004+\dots ,a_1=0.4,r=0.1$

$S_{\infty }=\frac{0.4}{1-0.1}=\frac{0.4}{0.9}=\frac{4}{9}$

---

4.5.4 Real-World Applications

Bouncing Ball Problem

A ball is dropped from height $H$ and rebounds to a fraction $r$ of its previous height each time ($0<r<1$).

• Initial drop: $H$ (downward only)
• Each subsequent bounce: the ball travels the rebound height twice (up then down)

$\text{Total distance}=H+2(rH+r^{2}H+r^{3}H+\dots )=H+\frac{2rH}{1-r}=H\cdot \frac{1+r}{1-r}$

Example 5

A ball is dropped from $10$ m and rebounds to $60\%$ of its previous height. Find the total distance.

Solution: $H=10$, $r=0.6$

$\text{Total distance}=10\cdot \frac{1+0.6}{1-0.6}=10\cdot \frac{1.6}{0.4}=40\text{ m}$

Rising Balloon / Investment Growth

When a quantity increases by a fixed fraction each period, the total accumulation over infinite periods is modeled by $S_{\infty }=\frac{a_1}{1-r}$, provided $|r|<1$.

---

Summary of Key Formulas

---