Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} n x^{n} $$

Step-by-Step Solution

Answer

The convergence set is \( |x| < 1 \).

Step 1: Apply the Ratio Test

The Ratio Test is a useful tool to determine the convergence of a power series. For the series \( \sum_{n=1}^{\infty} n x^{n} \), define \( a_n = n x^n \). We need to find \( \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \).

Step 2: Calculate Ratio for Consecutive Terms

Calculate \( \frac{a_{n+1}}{a_n} = \frac{(n+1)x^{n+1}}{n x^n} = \frac{n+1}{n} \cdot x = \left(1 + \frac{1}{n} \right)x \).

Step 3: Take the Limit

Find the limit as \( n \to \infty \) of \( \left| \frac{n+1}{n} x \right| \). This becomes \( \lim_{n \to \infty} \left| \left(1 + \frac{1}{n} \right)x \right| = |x| \).

Step 4: Apply the Ratio Test Conclusion

According to the Ratio Test, the series converges if \( |x| < 1 \) and diverges if \( |x| > 1 \). At \( |x| = 1 \), further investigation is needed, but generally, the series diverges since it behaves like a harmonic series.

Step 5: Determine the Convergence Set

Based on the Ratio Test and reasoning, the interval of convergence for the series is \( |x| < 1 \). The endpoints \( x = 1 \) and \( x = -1 \) result in divergent series.

Key Concepts

Ratio Test

The Ratio Test is an essential method in determining the convergence or divergence of a power series. When you have a series like \( \sum_{n=1}^{\infty} a_n \), you analyze the limit of the absolute value of the ratio of consecutive terms. Specifically, you evaluate \( \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \). The insights from this limit tell us:

• The series converges absolutely if the limit is less than 1.
• The series diverges if the limit is greater than 1.
• If the limit equals 1, the test is inconclusive.

For the power series \( \sum_{n=1}^{\infty} n x^{n} \), using the Ratio Test involves computing \( \left| \frac{a_{n+1}}{a_n} \right| \) resulting in \( |x| \), helping us analyze its convergence behavior.

interval of convergence

The interval of convergence refers to the set of all values of \( x \) for which a power series converges. When employing the Ratio Test on the series \( \sum_{n=1}^{\infty} n x^{n} \), we identify that the series converges when the result \( |x| < 1 \). This implies that:

• The series converges within the interval \(-1 < x < 1\).
• For this specific series, the endpoints \( x = 1 \) and \( x = -1 \) must be checked separately since the Ratio Test is inconclusive at these points.

The analysis shows that outside this interval, specifically at \( x = \pm 1 \), the series behaves like a harmonic series and generally diverges.

power series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n \) are constants and \( x \) is a variable. Such series expand functions and evaluate them for different values of \( x \). They can converge for certain intervals of \( x \) and diverge for others.

Power series are important because they allow us to represent functions as infinite polynomials. The interval of convergence is vital in determining where the power series representation is valid. In the exercise, \( \sum_{n=1}^{\infty} n x^{n} \), it is a power series with terms dependent on both \( n \) and \( x \), showcasing how the series evolves as \( n \) elevates indefinitely.

harmonic series

The harmonic series is an important series in mathematics, typically expressed as \( \sum_{n=1}^{\infty} \frac{1}{n} \). It is a classic example of a divergent series, as it adds terms that decrease in size but still result in an infinite sum.

In the context of the exercise, the behavior of the power series \( \sum_{n=1}^{\infty} n x^{n} \) at its endpoints \( x = \pm 1 \) can be linked to the harmonic series. Just like the harmonic series, these particular sums diverge. This relation emphasizes the intricacies of handling series at their convergence boundaries.