Problem 91 - Thomas Calculus

Question

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 series when it converges.

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern

Notice the infinite series alternates between terms of the form \( r^n \) and \( 2r^n \). The series is split into two series: one with terms \( r^{2n} \) and another with \( 2r^{2n+1} \).

Step 2: Analyze Subseries Summation Formulas

The series has two parts: \( 1 + r^2 + r^4 + \ldots \) and \( 2r + 2r^3 + 2r^5 + \ldots \). These correspond to geometric series with first term 1 and common ratio \( r^2 \), and first term \( 2r \) and common ratio \( r^2 \), respectively.

Step 3: Convergence Condition

For a geometric series \( a + ar + ar^2 + \ldots \) to converge, it must be that \(|r| < 1\). Thus, both subseries will converge if \(|r^2| < 1\), which implies \(|r| < 1\).

Step 4: Sum of Convergent Subseries

The sum of the first subseries \( 1 + r^2 + r^4 + \ldots \) is \( \frac{1}{1 - r^2} \). The sum of the second subseries \( 2r + 2r^3 + 2r^5 + \ldots \) is \( \frac{2r}{1 - r^2} \).

Step 5: Combine the Subseries

The sum of the entire series is the sum of these two separate series: \( \frac{1}{1 - r^2} + \frac{2r}{1 - r^2} = \frac{1 + 2r}{1 - r^2} \). The final convergent sum is \( \frac{1 + 2r}{1 - r^2} \).

Key Concepts

Series Convergence

In mathematics, convergence is a key concept when dealing with series and sequences. A series is said to be convergent if its terms approach a specific limiting value as you continue to add more terms. Convergence is particularly important for infinite series, since without convergence, you cannot sensibly talk about their sum. For a geometric series with a common ratio of \(r\), the series converges if the absolute value of the ratio \(|r|\) is less than 1. This is because when \(|r| < 1\), each successive term becomes smaller and closer to zero, allowing the sum to approach a finite number. In the context of our problem, the infinite series in question converges if the condition \(|r| < 1\) is met. This ensures that the behavior of the series stabilizes, and a finite sum can be computed.

Infinite Series

An infinite series is a sum of infinitely many terms. These series often arise in mathematics and require special consideration to determine if a meaningful sum exists. Infinite series can be highly useful, especially in calculating complex functions, approximations, and evaluations. There are different types of infinite series and one of the most common is the geometric series, characterized by a constant ratio between successive terms. In our exercise, the series: - Consists of terms like \(r^n\) and \(2r^n\).- Is broken into two simpler geometric subseries.- Relies on finding separate conditions for each subseries, both requiring \(|r|<1\) to converge, ultimately reflecting the behavior of the whole series.

Sum of Series

Calculating the sum of a series is frequently the ultimate goal, especially for convergent series. For geometric series, this summation can be straightforward, using the series sum formula. The sum formula for an infinite geometric series \(a + ar + ar^2 + \ldots\) is given by: \[\frac{a}{1-r}\]where \(a\) is the first term and \(r\) the common ratio.For the given series, it splits into two subseries with their respective sums:

• \( 1 + r^2 + r^4 + \ldots \) summing to \( \frac{1}{1 - r^2} \)
• \( 2r + 2r^3 + 2r^5 + \ldots \) summing to \( \frac{2r}{1 - r^2} \)

The sum of the overall series is thus:\[\frac{1 + 2r}{1 - r^2}\]indicating how the components of the series individually contribute to the sum.