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Chapter 9: Problem 91

For what values of \(r\) does the infinite series $$1+2 r+r^{2}+2 r^{3}+r^{4}+2 r^{5}+r^{6}+\cdots$$ converge? Find the sum of the serics when it converges.

Short Answer

Expert verified
The series converges for \(|r| < 1\) and the sum is \(\frac{1+2r}{1-r^2}\).

Step by step solution

01

Identify the Pattern of the Series

Observe the given series: \(1 + 2r + r^2 + 2r^3 + r^4 + 2r^5 + r^6 + \cdots\). Notice that the coefficients follow a pattern between 1 and 2, meaning each odd power of \(r\) has a coefficient of 2, while each even power of \(r\) has a coefficient of 1.
02

Decompose the Series into Two Separate Series

We can write the original series as the sum of two separate series:\[ (1 + r^2 + r^4 + r^6 + \, \cdots\,) + (2r + 2r^3 + 2r^5 + \, \cdots\,) \]The first series \(1 + r^2 + r^4 + \, \cdots\) is a geometric series with the first term 1 and common ratio \(r^2\). The second series \(2r + 2r^3 + 2r^5 + \, \cdots\) is also a geometric series with the first term \(2r\) and common ratio \(r^2\).
03

Determine the Convergence Condition

For both geometric series to converge, their common ratio must satisfy \(|r^2| < 1\). Therefore, \(|r| < 1\) is the condition for convergence.
04

Calculate the Sum of Each Series

Each geometric series can be summed using the formula \( \frac{a}{1 - r} \):- The sum of the first series is \( \frac{1}{1 - r^2} \) because the first term \(a = 1\) and the common ratio \(r = r^2\).- The sum of the second series is \( \frac{2r}{1 - r^2} \) because the first term \(a = 2r\) and the common ratio \(r = r^2\).
05

Combine the Sums

Add the sums of the two series to obtain the total sum of the original series:\[\frac{1}{1 - r^2} + \frac{2r}{1 - r^2} = \frac{1 + 2r}{1 - r^2}\]This is the sum of the original series, valid for \(|r| < 1\).

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Key Concepts

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

Infinite Series
An infinite series is a sum of an infinite sequence of terms. It can look like an endless list of numbers being added together. For example, in our exercise, the series is expressed as \(1 + 2r + r^2 + 2r^3 + r^4 + 2r^5 + r^6 + \cdots\). This means the sum continues indefinitely with no end.
A prime aspect of infinite series is that they can sometimes be simplified to give meaningful results despite having an infinite number of terms. In mathematics, we often look to understand whether an infinite series can converge to a specific value. If it does converge, it means we can calculate a precise sum for the series.
Convergence
Convergence of a series is when the sum of an infinite series adds up to a specific, finite number. Not all infinite series converge; some grow indefinitely. To solve the exercise, we needed to determine when the series converges.
For a geometric series, convergence occurs when the absolute value of the common ratio is less than 1, which means each term is progressively smaller, allowing the sum to settle at a certain number.
  • In our example, the series was broken into two geometric series.
  • Here, convergence required that the condition \(|r| < 1\) be satisfied for the whole series.
Ensuring this condition is met ensures that the whole original series converges to a finite sum.
Sum of Series
The sum of an infinite geometric series can be calculated if it converges.
If you have a series where each term reduces at a consistent rate, like the one with a pattern or ratio, you have a geometric series.
  • For the first series from our exercise, the sum is calculated as \( \frac{1}{1 - r^2} \), where 1 is the first term and \(r^2\) is the common ratio.
  • The second series yields a sum of \( \frac{2r}{1 - r^2} \), owing to its first term of \(2r\) with the same ratio \(r^2\).
You can see that even infinite series can have specific sums, provided they meet conditions for convergence.
Geometric Progression
A geometric series relies on the concept of geometric progression, where each term is multiplied by a fixed, common ratio to get to the next term.
For example, the series \(1, r^2, r^4, \ldots\) in the exercise is defined as a geometric progression because each term is obtained by multiplying the previous term by \(r^2\).
  • The other part \(2r, 2r^3, 2r^5, \ldots\) behaves similarly but starts at \(2r\).
  • To identify a geometric progression, look for this steady multiplication pattern from term to term.
Understanding geometric progression helps in spotting quickly whether an infinite series you encounter can converge and thus be safely summed.

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Most popular questions from this chapter

For a sequence \(\left\\{a_{n}\right\\}\) the terms of even index are denoted by \(a_{2 k}\) and the terms of odd index by \(a_{2 k+1} .\) Prove that if \(a_{2 k} \rightarrow L\) and \(a_{2 k+1} \rightarrow L,\) then \(a_{n} \rightarrow L\).

If \(\Sigma a_{n}\) is a convergent series of positive terms, prove that \(\Sigma \sin \left(a_{n}\right)\) converges.

A patient takes a 300 mg tablet for the control of high blood pressure every morning at the same time. The concentration of the drug in the patient's system decays exponentially at a constant hourly rate of \(k=0.12\) a. How many milligrams of the drug are in the patient's system just before the second tablet is taken? Just before the third tablet is taken? b. In the long run, after taking the medication for at least six months, what quantity of drug is in the patient's body just before taking the next regularly scheduled morning tablet?

It is not yet known whether the series $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \sin ^{2} n} $$converges or diverges. Use a CAS to explore the behavior of the series by performing the following steps. a. Define the sequence of partial sums $$ s_{k}=\sum_{n=1}^{k} \frac{1}{n^{3} \sin ^{2} n} $$ What happens when you try to find the limit of \(s_{k}\) as \(k \rightarrow \infty ?\) Does your CAS find a closed form answer for this limit? b. Plot the first 100 points \(\left(k, s_{k}\right)\) for the sequence of partial sums. Do they appear to converge? What would you estimate the limit to be? c. Next plot the first 200 points \(\left(k, s_{k}\right) .\) Discuss the behavior in your own words. d. Plot the first 400 points \(\left(k, s_{k}\right) .\) What happens when \(k=355 ?\) Calculate the number \(355 / 113 .\) Explain from you calculation what happened at \(k=355 .\) For what values of \(k\) would you guess this behavior might occur again?

Which of the series in Exercises \(55-62\) converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{n^{n}}{\left(2^{n}\right)^{2}}$$
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