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Exercises Sequences, counting and probability

Example 6 from Algebra and Trigonometry, 13.4 Series and Their Notations

Determining Whether the Sum of an Infinite Series is Defined

Determine whether the sum of each infinite series is defined.

ⓐ $12 + 8 + 4 + \dots$
ⓑ $\frac{3}{4} + \frac{1}{2} + \frac{1}{3} + ...$
ⓒ $\sum_{k = 1}^{\infty} 27 \cdot {(\frac{1}{3})}^{k}$
ⓓ $\sum_{k = 1}^{\infty} 5 k$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐThe ratio of the second term to the first is $\frac{2}{3},$ which is not the same as the ratio of the third term to the second, $\frac{1}{2} .$ The series is not geometric.
2. ⓑThe ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of $\frac{2}{3} .$ The sum of the infinite series is defined.
3. ⓒThe given formula is exponential with a base of $\frac{1}{3};$ the series is geometric with a common ratio of $\frac{1}{3} .$ The sum of the infinite series is defined.
4. ⓓThe given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

$\frac{1}{3} + \frac{1}{2} + \frac{3}{4} + \frac{9}{8} + ...$

How did it go?

PreviousSolving an Application Problem with a Geometric Series Next Finding the Sum of an Infinite Geometric Series

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 13.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.