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Chapter 7: Problem 64

Test for convergence or divergence, using each test at least once. Identify which test was used. (a) \(n\) th-Term Test (b) Geometric Series Test (c) \(p\) -Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test $$ \sum_{n=0}^{\infty} 5\left(-\frac{1}{5}\right)^{n} $$

Short Answer

Expert verified
The given series is a geometric series that satisfies the conditions for convergence. The sum of the series is \(4\)

Step by step solution

01

Identify Type of Series

Given series \(\sum_{n=0}^{\infty} 5\left(-\frac{1}{5}\right)^{n}\) is a Geometric series. A geometric series takes the form \(\sum ar^{n}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
02

Apply Geometric Series Test

The given series has the first term \(a = 5\) and common ratio \(r = -\frac{1}{5}\). A geometric series \(\sum ar^{n}\) converges if \(-1 < r < 1\). For this series, \(-1 < -\frac{1}{5} < 1\), so the Geometric Series Test can be assumed.
03

Determine Convergence or Divergence

Since the absolute value of the common ratio is less than 1, the series converges. For a converging geometric series, the sum is equal to \(\frac{a}{1 - r}\). Substituting \(a = 5\) and \(r = -\frac{1}{5}\) into this formula, we find that the sum of the series is equal to \(\frac{5}{1 - (-\frac{1}{5})} = 4\).

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Most popular questions from this chapter

(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{81} $$

The Fibonacci sequence is defined recursively by \(a_{n+2}=a_{n}+a_{n+1},\) where \(a_{1}=1\) and \(a_{2}=1\) (a) Show that \(\frac{1}{a_{n+1} a_{n+3}}=\frac{1}{a_{n+1} a_{n+2}}-\frac{1}{a_{n+2} a_{n+3}}\). (b) Show that \(\sum_{n=0}^{\infty} \frac{1}{a_{n+1} a_{n+3}}=1\).

Give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.) A monotonically increasing bounded sequence that does not converge

An electronic games manufacturer producing a new product estimates the annual sales to be 8000 units. Each year, \(10 \%\) of the units that have been sold will become inoperative. So, 8000 units will be in use after 1 year, \([8000+0.9(8000)]\) units will be in use after 2 years, and so on. How many units will be in use after \(n\) years?

Modeling Data The annual sales \(a_{n}\) (in millions of dollars) for Avon Products, Inc. from 1993 through 2002 are given below as ordered pairs of the form \(\left(n, a_{n}\right),\) where \(n\) represents the year, with \(n=3\) corresponding to 1993. (Source: 2002 Avon Products, Inc. Annual Report) (3,3844),(4,4267),(5,4492),(6,4814),(7,5079) (8,5213),(9,5289),(10,5682),(11,5958),(12,6171) (a) Use the regression capabilities of a graphing utility to find a model of the form \(a_{n}=b n+c, \quad n=3,4, \ldots, 12\) for the data. Graphically compare the points and the model. (b) Use the model to predict sales in the year 2008 .
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