Infinite Series

A geometric series is a series where the ratio between successive terms is constant. You can view a geometric series as a series with terms that form a geometric sequence (see the previous module on sequences).

For example, the series

$\sum_{i=0}^\infty\left(\frac{1}{3}\right)^i=1+\frac13+\frac19+\frac{1}{27}+\ldots$ is geometric with ratio

$r=\frac13$ .

Geometric series are our favorite series. It is always possible, and even easy, to determine whether a geometric series converges, and if it does, its value. This is rarely possible with other types of series.

The geometric series

$\displaystyle\sum_{i=0}^\infty a r^i=a+ar+ar^2+ar^3 + \ldots$ converges to

$\displaystyle\frac{a}{1-r}$ if

$-1<r<1$ and diverges otherwise. Warning: this value of the series is true only when the series begins with

$i=0$ , so that the first term is

$a$ .

We could also say a geometric series

$\sum ar^i$ converges if

$|r|<1$ which is the same as

$-1<r<1$ . The absolute value inequality

$|r|>1$ is equivalent to

$r<-1$ or

$r>1$ .

This video will use the

$n^{th}$ partial sums to show why we know the covergence and the value of geometric series.

The main issue is to confirm that a series is indeed geometric, and if so, what the values of $a$ and $r$ are.

Example: $5-\tfrac {10}{3}+\tfrac{20} {9}-\tfrac{40}{27}+\tfrac{80}{81}+\cdots.$

Solution: First, we try to determine if it is a geometric series. Dividing each term by the previous term shows that we have a common ratio of $r=-\tfrac 2 3$ , with $a=a\cdot r^0=5$ . So we have a geometric series, and since $|r|<1$ , the series converges. Its value is $\frac{a}{1-r}=\frac{5}{1+\tfrac 2 3}=\frac{5}{\tfrac 5 3}=3$ .
We have determined that $\displaystyle\sum_{n=0}^\infty 5\left(-\frac{2}{3}\right)^n=3$ .

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Example: $3-\tfrac 6 5 + \tfrac{12}{25}-\tfrac{24}{125}+\cdots$ .

DO: Find $r$ and $a$ , and evaluate this series.

Solution: $\displaystyle\sum_{n=0}^\infty3\cdot\left(-\frac{2}{5}\right)^n=\frac{15}{7}$ .

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Geometric Series