Problem 32 - Thomas Calculus

Question

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=1}^{\infty} \frac{(3 x+1)^{n+1}}{2 n+2} $$

Step-by-Step Solution

Answer

The series converges absolutely and conditionally for \(-\frac{2}{3} < x < 0\).

Step 1: Identify the series and its form

The given series is \( \sum_{n=1}^{\infty} \frac{(3x+1)^{n+1}}{2n+2} \). It can be identified as a power series in terms of \( (3x+1) \).

Step 2: Convert series to power series form

Re-write the series in the form \( \sum_{n=1}^{\infty} c_n (x-a)^n \) to analyze the radius of convergence.This series can be written as \( \sum_{n=1}^{\infty} \frac{(3x+1)^{n+1}}{2n+2} = \sum_{n=1}^{\infty} \frac{(3x+1)^{n}}{2n+2} \cdot (3x+1) \).

Step 3: Apply the Ratio Test for convergence

To find the radius of convergence, apply the Ratio Test:\[\lim_{n \to \infty} \left| \frac{(3x+1)^{n+1}}{2(n+1)+2} \cdot \frac{2n+2}{(3x+1)^n} \right|\]This simplifies to:\[\lim_{n \to \infty} \left| \frac{3x+1}{1 + \frac{2}{n}} \right| = |3x+1|\]Set this condition to be less than 1: \(|3x+1| < 1\).

Step 4: Solve the inequality for radius and interval

From \(|3x+1| < 1\), solve for \(x\):\[-1 < 3x + 1 < 1\]Subtract 1 from all parts to yield:\[-2 < 3x < 0\]Divide each term by 3:\[-\frac{2}{3} < x < 0\]The radius of convergence \( R \) is \( \frac{1}{3} \). Therefore, the interval of convergence is \((-\frac{2}{3}, 0)\).

Step 5: Check endpoints for absolute convergence

Test \(x = -\frac{2}{3}\):Substitute \(x = -\frac{2}{3}\) into the series: \[ \sum_{n=1}^{\infty} \frac{(3(-\frac{2}{3})+1)^{n+1}}{2n+2} = \sum_{n=1}^{\infty} 0\]No convergence testing needed as it evaluates to zero. Test \(x = 0\):Substitute \(x = 0\) into the series: \[ \sum_{n=1}^{\infty} \frac{1^{n+1}}{2n+2} = \sum_{n=1}^{\infty} \frac{1}{2n+2} \]This is a divergent harmonic series (alternating series).

Step 6: Conclusion on convergence

For this series, absolute convergence conditions match those for conditional convergence, as the series reduces at endpoints.

Key Concepts

series convergence

Series convergence is a fundamental concept in calculus and analysis. It refers to whether a series, which is an infinite sum of terms, adds up to a finite number. Understanding convergence is crucial because it helps determine when a series makes sense to sum up all of its terms.

Series can either converge or diverge:

• A convergent series has a sum that approaches a specific value as more terms are added.
• A divergent series doesn't settle at any particular value.

To analyze a series, we often check for absolute or conditional convergence. Absolute convergence means the series converges even when all terms are made positive. Conditional convergence suggests that it converges only when terms keep their signs. Recognizing convergence type is essential especially in alternating series, where terms alternate between positive and negative values.

In the given problem, convergence is explored first generally by transforming it into a more manageable power series form and then applying checks such as the Ratio Test to find its characteristics.

power series

A power series is a series in the form of \[\sum_{n=0}^{\infty} c_n(x-a)^n\]where each term consists of a coefficient \(c_n\), a variable \(x\), and a constant \(a\), raised to the power of \(n\). Such a series is centered around \(a\), and it can be used to represent functions in calculus.

Power series are very flexible:

• They represent many functions, like polynomials, by combining series terms.
• They simplify function calculations by providing series of terms centered at a point.

In the exercise provided, the given series transforms into a power series in terms of \((3x+1)\). This transformation helps in applying convergence tests easily.

The radius of convergence is the distance from the center \(a\) where this series continues to converge. It's a tangible measure and can affect where and how functions modeled by the series behave.

ratio test

The Ratio Test is a popular method for determining the convergence of series, particularly power series. It involves looking at the ratio of consecutive terms to see how they behave as \(n\) approaches infinity. This is vital for understanding whether the series converges or diverges.

Here's how the Ratio Test works:

• Take the absolute value of the ratio of the \(n+1\)-th term to the \(n\)-th term of the series.
• Evaluate the limit of this absolute ratio as \(n\) approaches infinity.
• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, or if it doesn’t exist, the series diverges.

For the given problem, the Ratio Test is used to find the radius of convergence by simplifying the expression and setting its absolute value less than 1 (\(|3x+1| < 1\)), indicating the interval where the series converges. By solving this inequality, one determines the values of \(x\) that ensure convergence.