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Chapter 9: Problem 5

Does a geometric series always have a finite value?

Short Answer

Expert verified
Explain. Answer: No, a geometric series does not always have a finite value. It converges and has a finite value if and only if the common ratio 'r' lies between -1 and 1, that is, -1 < r < 1. If the common ratio is greater than or equal to 1 or less than or equal to -1, the series will diverge and will not have a finite sum.

Step by step solution

01

Definition of a Geometric Series

A geometric series is the sum of the terms of a geometric sequence. A geometric sequence is a set of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). A geometric series can be represented as: S_n = a_1 + a_1 * r + a_1 * r^2 + a_1 * r^3 + ... + a_1 * r^(n-1), where n is the number of terms, and a_1 is the first term.
02

Convergence of a Geometric Series

A geometric series converges (i.e., has a finite value) if the sum of its terms approaches a finite limit as the number of terms increases. The convergence of a geometric series depends on the value of the common ratio 'r.' For a geometric series to converge, the common ratio 'r' must lie between -1 and 1, i.e., -1 < r < 1.
03

Formula for the Sum of a Convergent Geometric Series

If a geometric series converges, we can find the sum using the formula: S = a_1 * (1 - r^n) / (1 - r). In this formula, S represents the sum of the series, n is the number of terms, a_1 is the first term, and r is the common ratio. Notice that this formula is valid only when -1 < r < 1.
04

Divergence of a Geometric Series

If the common ratio 'r' is greater than or equal to 1, or less than or equal to -1, the geometric series will diverge, which means it will not have a finite sum. In this case, the series will either increase without bound or oscillate between two values.
05

Conclusion

A geometric series does not always have a finite value. It converges and has a finite value if and only if the common ratio 'r' lies between -1 and 1, i.e., -1 < r < 1. If the common ratio is greater than or equal to 1 or less than or equal to -1, the series will diverge and will not have a finite sum.

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Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{\cos k}{k^{3}}$$

Determine whether the following statements are true and give an explanation or counterexample. a. A series that converges must converge absolutely. b. A series that converges absolutely must converge. c. A series that converges conditionally must converge. d. If \(\sum a_{k}\) diverges, then \(\Sigma\left|a_{k}\right|\) diverges. e. If \(\sum a_{k}^{2}\) converges, then \(\sum a_{k}\) converges. f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\Sigma a_{k}^{2}\) converges. g. If \(\Sigma a_{k}\) converges conditionally, then \(\Sigma\left|a_{k}\right|\) diverges.

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Consider the expression \(\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}},\) where the process continues indefinitely. a. Show that this expression can be built in steps using. the recurrence relation \(a_{0}=1, a_{n+1}=\sqrt{1+a_{n}},\) for \(n=0,1,2,3, \ldots .\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Estimate the limit of the sequence. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. e. Repeat the preceding analysis for the expression \(\sqrt{p+\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}}}\) where \(p > 0 .\) Make a table showing the approximate value of this expression for various values of \(p .\) Does the expression seem to have a limit for all positive values of \(p ?\)
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