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Chapter 9: Problem 49

Explain what is wrong with the statement. The sum of the infinite geometric series \(1-\frac{3}{2}+\frac{9}{4}-\) \(\frac{27}{8}+\dots\) is \(\frac{1}{1+3 / 2}=\frac{2}{5}\)

Short Answer

Expert verified
The series does not converge because the common ratio \(|r| = 1.5 > 1\).

Step by step solution

01

Identify the Series

The given series is an infinite geometric series: \(1 - \frac{3}{2} + \frac{9}{4} - \frac{27}{8} + \dots\). The first term \(a = 1\) and the common ratio \(r\) can be found by dividing the second term by the first term, \(r = -\frac{3}{2}\).
02

Determine the Convergence Criterion

For an infinite geometric series to converge and have a sum, the absolute value of the common ratio must be less than 1, i.e. \(|r| < 1\). Here, \(r = -\frac{3}{2}\), so \(|r| = \frac{3}{2} = 1.5\).
03

Check if the Series Converges

Since \(|r| = 1.5 > 1\), the series does not converge. Therefore, it does not have a finite sum.
04

Analyze the Error in the Statement

The mistake in the statement is assuming the series converges and applying the sum formula \(S = \frac{a}{1 - r}\), which is valid only for \(|r| < 1\). Calculating \(\frac{1}{1 + \frac{3}{2}}\) is not appropriate here since the series does not meet the convergence criteria.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
In an infinite geometric series, the common ratio, denoted by \(r\), is an essential element. It determines the pattern in which the terms of the series increase or decrease. You can find the common ratio by dividing any subsequent term by its preceding term, such as \(r = \frac{-\frac{3}{2}}{1} = -\frac{3}{2}\).
- **Why is it Important?** - **Defines the Series Pattern:** The common ratio tells us how each term is related to the previous one. - **Impacts Convergence:** Whether the series converges or not largely depends on the value of \(r\).
If \(|r| < 1\), the terms get progressively smaller in magnitude, potentially allowing the series to converge. In contrast, if \(|r|\) is greater or equal to 1, the terms do not diminish or may even increase, preventing convergence.
Convergence Criterion
The convergence criterion is a rule to determine if an infinite geometric series will converge to a finite sum.
- **Key Rule:** A geometric series converges only if the absolute value of the common ratio is less than 1, expressed as \(|r| < 1\). In our example, since \(r = -\frac{3}{2}\), we find that \(|r| = 1.5\), which certainly does not satisfy this criterion, as it is greater than 1.
- **What happens when \(|r|\) is not less than 1?** - The series fails to converge as the terms don't steadily approach zero. - The partial sums can grow indefinitely, meaning there's no single sum for the infinite series.
Understanding this criterion protects against incorrectly applying formulas only appropriate for convergent series.
Sum Formula for Geometric Series
Finding the sum of an infinite geometric series is straightforward if it meets the convergence criteria. The formula is given by, \[ S = \frac{a}{1 - r} \] where \(a\) is the first term and \(r\) is the common ratio.
  • Applicable only when \(|r| < 1\): Validity of this formula hinges on fulfilling the convergence criterion. Otherwise, using it would lead to erroneous conclusions.
  • Mistake in the Example: In our example series, attempting to calculate the sum with \(|r| = 1.5\) leads to inaccuracies since the series cannot sum to a finite value.
Understanding when and how to use this formula is crucial to correctly solving problems involving infinite geometric series. Recognizing when a series converges ensures proper application of the sum formula for accurate results.

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Most popular questions from this chapter

Introduce the root test for convergence. Given a series \(\sum a_{n}\) of positive terms (that is, \(a_{n}>0\) ) such that the root \(\sqrt[n]{a_{n}}\) has a limit \(r\) as \(n \rightarrow \infty\) \(\cdot\) if \(r<1,\) then \(\sum a_{n}\) converges if \(r>1,\) then \(\sum a_{n}\) diverges if \(r=1,\) then \(\sum a_{n}\) could converge or diverge. (This test works since \(\lim _{n \rightarrow \infty} \sqrt[n]{a_{n}}=r\) tells us that the series is comparable to a geometric series with ratio \(r .\) ) Use this test to determine the behavior of the series.$$\sum_{n=1}^{\infty}\left(\frac{5 n+1}{3 n^{2}}\right)^{n}$$.

A series \(\sum a_{n}\) of positive terms (that is, \(a_{n}>0\) ) can be used to form another series \(\sum b_{n}\) where each term \(b_{n}\) is the average of the first \(n\) terms of the original series, that is, \(b_{n}=\left(a_{1}+a_{2}+\cdots+a_{n}\right) / n .\) Show that \(\sum b_{n}\) does not converge (even if \(\sum a_{n}\) does). [Hint: Compare \(\sum b_{n}\) to a multiple of the harmonic series.]

What values of \(a\) does the series converge? $$\sum_{n=1}^{\infty}(\ln a)^{n}, a>0 \quad$$

Use a computer or calculator to investigate the behavior of the partial sums of the alternating series. Which appear to converge? Confirm convergence using the alternating series test. If a series converges, estimate its sum.$$1-0.1+0.01-0.001+\cdots+(-1)^{n} 10^{-n}+\cdots$$

The Fibonacci sequence, first studied by the thirteenth century Italian mathematician Leonardo di Pisa, also known as Fibonacci, is defined recursively by \(F_{n}=F_{n-1}+F_{n-2}\) for \(n>2\) and \(F_{1}=1, F_{2}=1\) The Fibonacci sequence occurs in many branches of mathematics and can be found in patterns of plant growth (phyllotaxis). (a) Find the first 12 terms. (b) Show that the sequence of successive ratios \(F_{n+1} / F_{n}\) appears to converge to a number \(r\) satisfying the equation \(r^{2}=r+1 .\) (The number \(r\) was known as the golden ratio to the ancient Greeks.) (c) Let \(r\) satisfy \(r^{2}=r+1 .\) Show that the sequence \(s_{n}=A r^{n},\) where \(A\) is constant, satisfies the Fibonacci equation \(s_{n}=s_{n-1}+s_{n-2}\) for \(n>2\)
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