Question

Determine the radius of convergence of the given power series. $$ \sum_{n=1}^{\infty} \frac{(2 x+1)^{n}}{n^{2}} $$

Short answer

The radius of convergence of the given power series is 1. This means that the power series converges absolutely for all values of x within the interval (-1, 0).

Step-by-step solution

Write down the Ratio Test formula

For a power series \(\sum_{n=1}^{\infty} a_n\), the Ratio Test states that the limit: $$ L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| $$ helps determine the convergence of the series. If \(L < 1\), the series converges. If \(L > 1\), the series diverges. If \(L = 1\), the test is inconclusive.

Apply the Ratio Test to the given power series

For the given power series, the general term is \(a_n = \frac{(2x+1)^n}{n^2}\). We want to find the ratio \(\frac{a_{n+1}}{a_n}\) and take the limit as n approaches infinity. $$ \frac{a_{n+1}}{a_n} = \frac{\frac{(2x+1)^{n+1}}{(n+1)^2}}{\frac{(2x+1)^n}{n^2}} $$

Simplify the ratio

Now we can simplify the ratio, canceling out some terms: $$ \frac{a_{n+1}}{a_n} = \frac{(2x+1)^{n+1} n^2}{(n+1)^2 (2x+1)^n} $$ Since \((2x+1)^n\) appears in both the numerator and the denominator, we can cancel it out: $$ \frac{a_{n+1}}{a_n} = \frac{(2x+1) n^2}{(n+1)^2} $$

Take the limit as n approaches infinity

Now we need to take the limit of the simplified ratio as n approaches infinity: $$ L = \lim_{n \to \infty} \left|\frac{(2x+1) n^2}{(n+1)^2}\right| $$ To evaluate this limit, we can divide both the numerator and denominator by \(n^2\): $$ L = \lim_{n \to \infty} \left|\frac{(2x+1)}{\left(1+\frac{1}{n}\right)^2}\right| $$ As n approaches infinity, the term \(\frac{1}{n}\) becomes 0, giving us: $$ L = \left|\frac{(2x+1)}{1^2}\right| = |2x+1| $$

Find the radius of convergence

For the series to converge, we need the limit \(L < 1\). We have \(L = |2x+1|\), so: $$ |2x+1| < 1 $$ To find the radius of convergence, we need to solve this inequality for the absolute value of x: $$ -1 < 2x+1 < 1 $$ Subtracting 1 from all sides gives: $$ -2 < 2x < 0 $$ Dividing by 2: $$ -1 < x < 0 $$ Thus, the radius of convergence is: $$ R = |x| = 1 $$ So the radius of convergence of the given power series is 1.

Key concepts

Ratio Test

When it comes to analyzing the convergence of power series, the Ratio Test is an essential tool. It provides a way to determine whether a series converges absolutely, diverges, or remains inconclusive by using the limit of the ratio of successive terms. Here's a simplified explanation:

In the Ratio Test, we consider a series \( \sum_{n=1}^\infty a_n \) and calculate the limit \( L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| \) as n approaches infinity. If this limit, \( L \), is less than 1, the series is said to converge absolutely. If it's greater than 1, the series diverges. And if the limit equals 1, the test does not provide a conclusive answer on the convergence of the series.

To ensure students grasp this concept effectively:

• Highlight the significance of the absolute value in this test, ensuring that the ratio considered is non-negative.
• Provide examples where the Ratio Test clearly shows convergence or divergence.
• Discuss how the Ratio Test can be inconclusive, requiring other convergence tests to make a determination.

Power Series

A power series is a series of the form \( \sum_{n=0}^\infty a_n (x - c)^n \), where \( a_n \) represents the coefficients, \( x \) is the variable, and \( c \) is the center of the series. The series expands in powers of \( (x-c) \).

Power series are used extensively in fields like physics and engineering to approximate functions, solve differential equations, and model complex phenomena. To further enhance comprehension for students:

• Clarify the role of the center, \( c \), and how it affects the series.
• Discuss examples where the power series converges to well-known functions.
• Show the connection between Taylor series and power series for function approximation.

The application of the Ratio Test to a power series helps determine the radius of convergence, which indicates the interval around the center \( c \) where the series converges.

Limit of a Sequence

Understanding the limit of a sequence is fundamental in calculus and analysis. The limit describes the value that the terms of a sequence get closer to as the index \( n \) grows indefinitely. It is denoted as \( \lim_{n \to \infty} a_n \) for a sequence \( \{a_n\} \) and can be a finite number, infinity, or it may not exist at all.

For students to understand this concept standout points are:

• Explain the epsilon-N definition of limits, which provides an intuitive grasp of how close terms must get to the limit as \( n \) increases.
• Use graphical illustrations of sequences approaching their limits to visually demonstrate the concept.
• Highlight differences between limits that exist (finite or infinite) and those that do not, using diverse examples.

The calculation of the limit is crucial for the Ratio Test during the investigation of the radius of convergence for power series.