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Chapter 9: Problem 39

Evaluate the geometric series or state that it diverges. $$\sum_{k=2}^{\infty}(-0.15)^{k}$$

Short Answer

Expert verified
Answer: The sum of the infinite geometric series is approximately 0.019565.

Step by step solution

01

Identify the common ratio

The common ratio (r) is the factor between each term in the sequence. In our case, it's (-0.15).
02

Determine whether the series converges or diverges

To determine if the given series converges, we need to check if the common ratio lies between -1 and 1. $$-1 < -0.15 < 1$$ Since the common ratio lies between -1 and 1, the series converges.
03

Find the first term

The series starts from $$k=2$$, so the first term in the series (a) would be $$(-0.15)^2$$. $$a = (-0.15)^2 = 0.0225$$
04

Calculate the sum of the series

Since the series converges, we can use the formula for the sum of an infinite geometric series. $$S=\frac{a}{1-r}$$ Plug in the values \(a=0.0225\) and \(r=-0.15\): $$S=\frac{0.0225}{1-(-0.15)}$$ $$S=\frac{0.0225}{1.15}$$ $$S\approx0.019565$$ Therefore, the sum of the infinite geometric series $$\sum_{k=2}^{\infty}(-0.15)^{k}$$ is approximately 0.019565.

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Key Concepts

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

Convergence
In the context of geometric series, convergence refers to the series tending toward a finite sum as more and more terms are added to it. For a geometric series to converge, the absolute value of the common ratio must be less than 1. This means:
  • The common ratio, denoted as \( r \), lies within the interval \(-1 < r < 1\).
  • If \( |r| \geq 1 \), the series diverges, meaning it doesn't settle on any value but tends toward infinity or behaves erratically.
In our case, since the common ratio is \(-0.15\), which satisfies the condition \(-1 < -0.15 < 1\), the series converges. This allows us to find its sum using the appropriate formula. Understanding convergence is crucial for determining whether it is even possible to find a sum for an infinite series.
Infinite Series
An infinite series in mathematics is a sum of infinitely many terms. When evaluating such series, especially geometric ones, we are interested in whether the series forms a pattern that converges to a fixed value. The series given in the exercise is:
  • \( \sum_{k=2}^{\infty}(-0.15)^{k} \)
  • Starts from \( k=2 \), meaning the first term we'll consider is \((-0.15)^2\).
The sum of an infinite geometric series can only be found when the series converges. If it diverges, the sum is undefined. When convergent, the sum \( S \) of the series can be calculated using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term. This formula provides a finite sum, even though the series itself has infinitely many terms.
Common Ratio
The common ratio \( r \) in a geometric series is the constant factor by which each term is multiplied to get the next term. It plays a critical role in determining both the pattern and behavior of the series. Key characteristics include:
  • The common ratio is found by dividing any term by the previous term in the series.
  • In our example, the common ratio is \(-0.15\).
  • This value significantly dictates the convergence; if \(-1 < r < 1\), the series converges.
The common ratio directly affects how the terms grow or shrink as you progress through the series. When \( r \) is a negative number, terms will alternate in sign, leading to a potential cancelling effect in terms of sums. Knowing \( r \) helps us apply the infinite geometric series sum formula accurately upon confirming the series converges.

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Most popular questions from this chapter

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

For a positive real number \(p,\) how do you interpret \(p^{p^{p \cdot *}},\) where the tower of exponents continues indefinitely? As it stands, the expression is ambiguous. The tower could be built from the top or from the bottom; that is, it could be evaluated by the recurrence relations \(a_{n+1}=p^{a_{n}}\) (building from the bottom) or \(a_{n+1}=a_{n}^{p}(\text { building from the top })\) where \(a_{0}=p\) in either case. The two recurrence relations have very different behaviors that depend on the value of \(p\). a. Use computations with various values of \(p > 0\) to find the values of \(p\) such that the sequence defined by (2) has a limit. Estimate the maximum value of \(p\) for which the sequence has a limit. b. Show that the sequence defined by (1) has a limit for certain values of \(p\). Make a table showing the approximate value of the tower for various values of \(p .\) Estimate the maximum value of \(p\) for which the sequence has a limit.

It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the (conditionally convergent) alternating harmonic series has the value $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots=\ln 2.$$ Show that by rearranging the terms (so the sign pattern is \(++-\) ), $$1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\cdots=\frac{3}{2} \ln 2.$$

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=\sqrt{2+a_{n}} ; a_{0}=1, n=0,1,2, \dots$$

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year \(1984 .\) Assume the CPI has increased by an average of \(3 \%\) per year since \(1984 .\) Let \(c_{n}\) be the CPI \(n\) years after \(1984,\) where \(c_{0}=100.\)
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