TBIL-Cal Geometric Series (SQ4)

Section 8.4 Geometric Series (SQ4)

Learning Outcomes

Determine if a geometric series converges, and if so, the value it converges to.

Subsection 8.4.1 Activities

Activity 8.4.1.

Recall from Section 8.3 that for any real numbers

a, r

and

S_{n} = \sum_{i = 0}^{n} a r^{i}

that:

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\begin{aligned} S_{n} = \sum_{i = 0}^{n} a r^{i} & = a + a r + a r^{2} + \dots a r^{n} \\ (1 - r) S_{n} = (1 - r) \sum_{i = 0}^{n} a r^{i} & = (1 - r) (a + a r + a r^{2} + \dots a r^{n}) \\ (1 - r) S_{n} = (1 - r) \sum_{i = 0}^{n} a r^{i} & = a - a r^{n + 1} \\ S_{n} & = a \frac{1 - r^{n + 1}}{1 - r} . \end{aligned}

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(a)

Using Definition 8.3.12, for which values of

r

does

\sum_{n = 0}^{\infty} a r^{n}

converges?

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$| r | > 1 .$
$| r | = 1 .$
$| r | < 1 .$
The series converges for every value of $r .$

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(b)

Where possible, determine what value

\sum_{n = 0}^{\infty} a r^{n}

converges to.

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Fact 8.4.2.

Geometric series are sums of the form

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\sum_{n = 0}^{\infty} a r^{n} = a + a r + a r^{2} + a r^{3} + \dots,

where

a

and

r

are real numbers. When

| r | < 1

this series converges to the value

\frac{a}{1 - r} .

Otherwise, the geometric series diverges.

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Activity 8.4.3.

Consider the infinite series

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5 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \dots .

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(a)

Complete the following rearrangement of terms.

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\begin{aligned} 5 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \dots & = ? + (3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \dots) \\ = ? + \sum_{n = 0}^{\infty} ? \cdot {(\frac{1}{?})}^{n} \end{aligned}

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(b)

Since

| \frac{1}{?} | < 1,

this series converges. Use the formula

\sum_{n = 0}^{\infty} a r^{n} = \frac{a}{1 - r}

to find the value of this series.

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$\frac{7}{2}$
$\frac{13}{2}$
$8$
$10$

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Activity 8.4.4.

Complete the following calculation, noting

| 0.6 | < 1 :

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\begin{aligned} \sum_{n = 2}^{\infty} 2 (0.6)^{n} & = (\sum_{n = 0}^{\infty} 2 (0.6)^{n}) - ? - ? \\ = (\frac{?}{1 - ?}) - ? - ? \end{aligned}

What does this simplify to?

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$1.1$
$1.4$
$1.8$
$2.1$

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Observation 8.4.5.

Given a series that appears to be mostly geometric such as

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3 + (1.1)^{3} + (1.1)^{4} + \dots (1.1)^{n} + \dots

we can always rewrite it as the sum of a standard geometric series with some finite modification, in this case:

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- 0.31 + \sum_{n = 0}^{\infty} (1.1)^{n}

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Thus the original series converges if and only if

\sum_{n = 0}^{\infty} (1.1)^{n}

converges.

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When the series diverges as in this example, then the reason why (

| 1.1 | \geq 1

) can be seen without any modification of the original series.

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Activity 8.4.6.

For each of the following modified geometric series, determine without rewriting if they converge or diverge.

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(a)

- 7 + {(- \frac{3}{7})}^{2} + {(- \frac{3}{7})}^{3} + \dots .

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(b)

- 6 + {(\frac{5}{4})}^{3} + {(\frac{5}{4})}^{4} + \dots .

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(c)

4 + \sum_{n = 4}^{\infty} {(\frac{2}{3})}^{n} .

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(d)

8 - 1 + 1 - 1 + 1 - 1 + \dots .

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Activity 8.4.7.

Find the value of each of the following convergent series.

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(a)

- 1 + \sum_{n = 1}^{\infty} 2 \cdot {(\frac{1}{2})}^{n} .

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(b)

- 7 + {(- \frac{3}{7})}^{2} + {(- \frac{3}{7})}^{3} + \dots .

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(c)

4 + \sum_{n = 4}^{\infty} {(\frac{2}{3})}^{n} .

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Subsection 8.4.2 Videos

Figure 184. Video: Determine if a geometric series converges, and if so, the value it converges to.

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Subsection 8.4.3 Exercises

Exercises available at https://tbil.org/preview/calculus/exercises/#/bank/SQ4/

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