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Chapter 10: Problem 21

Find the sum, if it exists. $$-2+1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\cdots$$

Short Answer

Expert verified
The sum of the series is \(-\frac{7}{3}\).

Step by step solution

01

Identify the Pattern

Observe the sequence: \(-2 + 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\). Each term after the initial \(-2\) is a power of \(-\frac{1}{2}\), giving the general form \(a_n = \left( -\frac{1}{2} \right)^{n-1}\). Thus the series is \(-2 + \sum_{n=1}^{\infty} \left( -\frac{1}{2} \right)^{n}\).
02

Separate and Recognize the Series Type

Recognize the series \(\sum_{n=1}^{\infty} \left( -\frac{1}{2} \right)^n\) as a geometric series with the first term \(-\frac{1}{2}\) and common ratio \(r = -\frac{1}{2}\).The geometric series formula \(S = \frac{a}{1-r}\) applies when \(|r| < 1\).
03

Calculate the Sum of the Geometric Series

Use the formula to calculate the sum of the geometric series:\(a = -\frac{1}{2}\) and \(r = -\frac{1}{2}\), giving:\[ S = \frac{-\frac{1}{2}}{1 - (-\frac{1}{2})} = \frac{-\frac{1}{2}}{1 + \frac{1}{2}} = \frac{-\frac{1}{2}}{\frac{3}{2}} = -\frac{1}{3} \]
04

Combine the Results

Now combine the sum of the geometric series with the initial term \(-2\):\( S_{total} = -2 + (-\frac{1}{3}) = -2 - \frac{1}{3} = -\frac{6}{3} - \frac{1}{3} = -\frac{7}{3} \).
05

Verify the Convergence Conditions

Verify the common ratio \(r = -\frac{1}{2}\) satisfies \(|r| < 1\), ensuring convergence of the geometric series. Since it does, our calculation is valid.

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Key Concepts

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

Sum of Series
In the world of sequences and series, the sum of a series refers to the total value that the series converges to, if it does indeed converge. In this particular exercise, we are asked to find the sum of an infinite series that starts with
  • a distinct initial term, -2, and
  • a geometric sequence with a common ratio.
The sum of a geometric series can be found using the formula \( S = \frac{a}{1-r} \). This formula works for infinite geometric series under the condition \(|r| < 1\). Here,
  • \( a \) represents the first term of the geometric sequence (not of the entire series), and
  • \( r \) is the common ratio of the series.
After recognizing the series' pattern, we can calculate the sum of the geometric part separatelyand then add it to the initial term to get the total sum of the whole series.This step-by-step method ensures that we accurately account for all parts of the series.
Convergence
Convergence is a key consideration when dealing with infinite series. It indicates whether the series settles to a specific number (sum) as more and more terms are added.For a series to converge, it is crucial that its terms decrease and approach zero as we progress further into it. However, this alone doesn't guarantee convergence.With geometric series, convergence is determined largely by the common ratio \( r \). Important facts regarding convergence include:
  • A geometric series will converge only if \(|r| < 1\).This is because when \(|r| < 1\), the terms become smaller and smaller in magnitude, approaching zero.
  • If \(|r| \geq 1\), the series diverges, meaning it doesn't settle to a fixed sum.
In this exercise, because the common ratio is \(-\frac{1}{2}\), which indeed has \(|r| < 1\),the series is confirmed to converge.Thanks to this convergence,we can safely use the geometric series formula to find its sum.
Common Ratio
The common ratio \( r \) is an essential trait of any geometric series. It is the factor by which terms are multiplied to obtain the subsequent term. The common ratio gives the series its "multiplicative" characteristic. Here's how it relates to our exercise:
  • To determine the common ratio in a given series, we divide any term in the series by the term immediately preceding it.For example, using the sequence \( 1, -\frac{1}{2}, \frac{1}{4}, \ldots \),the common ratio is found one term divided by the previous one: \( r = \frac{-\frac{1}{2}}{1} = -\frac{1}{2} \).
  • The series can uniquely behave depending on \( r \):
    • If \(|r| < 1\), terms shrink,leading to convergence.
    • If \(|r| = 1\), terms maintain the same size, leading to potential oscillation.
    • If \(|r| > 1\), terms grow larger, leading to divergence.
Therefore, identifying the common ratio correctly is crucial not only for determining the series behavior but also for applying necessary mathematical formulas to determine its sum.

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Most popular questions from this chapter

Every year, a company sells 1000 units of a product while \(20 \%\) of the total number in use fail. Assume sales are at the start of the year and failures are at the end of the year. (a) Find the market stabilization point for this product. (b) If the stabilization point is approached very slowly, the number of units in use may not get close to this value because market conditions change first. Make a table for \(S_{n},\) the number of units in use right after the \(n^{\text {th }}\) annual sale, for \(n=5,10,15,20,\) to see how rapidly this market approaches the stabilization point.

We use \(1500 \mathrm{~kg}\) of a mineral this year and consumption of the mineral is increasing annually by \(4 \%\). The total reserves of the mineral are estimated to be \(120,000 \mathrm{~kg}\). Approximately when will the reserves run out?

(a) A dose \(D\) of a drug is administered at intervals equal to the half-life. (That is, the second dose is given when half the first dose remains.) At the steady state, find the quantity of drug in the body right after a dose. (b) If the quantity of a drug in the body after a dose is \(300 \mathrm{mg}\) at the steady state and if the interval between doses equals the half-life, what is the dose?

Every month, \(\$ 500\) is deposited into an account earning \(0.1 \%\) interest a month, compounded monthly. (a) How much is in the account right after the \(6^{\text {th }}\) deposit? Right before the \(6^{\text {th }}\) deposit? (b) How much is in the account right after the \(12^{\text {th }}\) deposit? Right before the \(12^{\text {th }}\) deposit?

Find the sum, if it exists. $$75+75(0.22)+75(0.22)^{2}+\cdots$$
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