Problem 3 - Differential Equations and Linear Algebra

Question

Determine the radius of convergence of the given power series. $$\sum_{n=0}^{\infty} \frac{x^{n}}{n^{2}}$$

Step-by-Step Solution

Answer

The radius of convergence for the given power series \(\sum_{n=0}^{\infty} \frac{x^{n}}{n^{2}}\) is \(r = 1\).

Step 1: Apply the Ratio Test

To find the radius of convergence for the given power series, we will use the Ratio Test. The Ratio Test states that the given series \(\sum_{n=0}^{\infty} a_n\) converges absolutely if the limit \[\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} < 1\] exists. In our case, the series is \(\sum_{n=0}^{\infty} \frac{x^{n}}{n^{2}}\), so the terms of the series are \(a_n = \frac{x^n}{n^2}\).

Step 2: Compute the limit for the Ratio Test

We will now compute the limit of the ratio test using the terms of the series: \[\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} = \lim_{n\to\infty} \frac{\left|\frac{x^{n+1}}{(n+1)^2}\right|}{\left|\frac{x^n}{n^2}\right|}\] First, we can simplify the expression: \[\lim_{n\to\infty}\frac{\frac{|x^{n+1}|}{|(n+1)^2|}}{\frac{|x^n|}{|n^2|}} = \lim_{n\to\infty}\frac{|x^{n+1}|n^2}{|x^n|(n+1)^2}\] Next, we simplify the powers of x and compute the limit: \[\lim_{n\to\infty}\frac{|x|n^2}{(n+1)^2} = \frac{|x|}{1}\lim_{n\to\infty}\frac{n^2}{(n+1)^2}\] Now, we have a rational function with power in the numerator and the denominator, so we can find the limit by dividing the numerator and the denominator by the highest power of n: \[\lim_{n\to\infty}\frac{n^2}{(n+1)^2} = \lim_{n\to\infty}\frac{1}{\left(1+\frac{1}{n}\right)^2} = \frac{1}{1^2} = 1\]

Step 3: Determine the radius of convergence

We have found that the limit for the ratio test is equal to \(\frac{|x|}{1}\), so the series converges absolutely when \(\frac{|x|}{1} < 1\). This inequality is equivalent to: \[|x| < 1\] Thus, the radius of convergence for the given power series is \(r = 1\).

Key Concepts

Power Series

A power series is a type of infinite series that takes the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n \) are coefficients and \( c \) is the center of the series. In simple terms, a power series is like a polynomial with infinitely many terms that goes on forever.

• The variable \( x \) in the series represents the input, and it is raised to a power that increases with each term.
• In the power series \( \sum_{n=0}^{\infty} \frac{x^n}{n^2} \), the center \( c \) is zero, and it simplifies to just \( x^n \) since there is no shift needed around a center.
• Each term in this series looks like \( \frac{x^n}{n^2} \), combining the variable power \( x^n \) with a coefficient of \( \frac{1}{n^2} \).

Understanding power series is crucial because they allow us to explore how different functions behave as we vary \( x \). In mathematical analysis, power series are extensively used to approximate functions around a particular point.

Ratio Test

The Ratio Test is an efficient tool to determine the convergence of a power series. It helps by examining the ratio of successive terms in the series.

• The basic idea is to take the absolute value of the next term divided by the previous term, \( \lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} \).
• If this limit is less than 1, then the series converges absolutely.
• For the series \( \sum_{n=0}^{\infty} \frac{x^n}{n^2} \), we found \( \lim_{n\to\infty} \frac{|x| n^2}{(n+1)^2} \).

We observed that as \( n \) becomes very large, the limit simplifies to \( \frac{|x|}{1}=|x| \). By setting \( |x| < 1 \), it implies that the Radius of Convergence, which marks the range of \( x \) for which the series sum will be accurate and finite, is 1.

Convergence of Series

The concept of convergence of a series is all about determining if an infinite series adds up to a fixed number.

• If a series converges, it means as you sum its terms, they approach a finite value.
• For power series, convergence depends heavily on the value of \( x \). This is where the radius of convergence becomes significant.
• For our specific power series \( \sum_{n=0}^{\infty} \frac{x^n}{n^2} \), the radius of convergence calculated was 1.

This means the series will converge whenever \( |x| < 1 \). However, exactly at \( |x| = 1 \), the test does not guarantee convergence. Within the radius of convergence, you can sum the series safely to get meaningful results. Outside this range, the series terms grow too large or behave improperly to sum up nicely.