Problem 6 - Differential Equations and Linear Algebra

Question

Determine the radius of convergence of the given power series. $$\sum_{n=0}^{\infty} \frac{5^{n} x^{n}}{n !}$$

Step-by-Step Solution

Answer

The radius of convergence for the given power series \(\sum_{n=0}^{\infty} \frac{5^{n} x^{n}}{n !}\) is infinity, as the power series converges for any value of x.

Step 1: Apply the Ratio Test

The Ratio Test is defined as \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\), where a_n represents the terms in the series. We can write the given power series as a_n = \(\frac{5^n x^n}{n!}\) and a_{n+1} = \(\frac{5^{n+1} x^{n+1}}{(n+1)!}\). Step 2: Compute the limit

Step 2: Compute the limit

Next, we need to compute the limit of the Ratio Test as n approaches infinity: \(\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\). We have \(\frac{a_{n+1}}{a_n} = \frac{ \frac{5^{n+1} x^{n+1}}{(n+1)!} }{ \frac{5^n x^n}{n!} } \). Step 3: Simplify the expression

Step 3: Simplify the expression

We now simplify the expression: \(\frac{ \frac{5^{n+1} x^{n+1}}{(n+1)!} }{ \frac{5^n x^n}{n!} } = \frac{5^{n+1}x^{n+1}}{(n+1)!} \cdot \frac{n!}{5^n x^n} = \frac{5x(n!)}{(n+1)!}\). Step 4: Compute the limit

Step 4: Compute the limit

Compute the limit as n approaches infinity: \(\lim_{n \to \infty} \left| \frac{5x(n!)}{(n+1)!} \right|\). Observe that (n+1)! is equivalent to (n+1)n!. The expression becomes: \(\lim_{n \to \infty} \left| \frac{5x}{n+1} \right|\). Step 5: Determine the criterion for convergence

Step 5: Determine the criterion for convergence

According to the Ratio Test, if the absolute value of the limit is less than 1, the series converges. In this case, we need to determine when \(\left| \frac{5x}{n+1} \right| < 1\). Since n tends to infinity, the limit is clearly 0, and 0 is less than 1. Therefore, the condition for convergence is met for all x. Step 6: State the radius of convergence

Step 6: State the radius of convergence

Since the series converges for all x, the radius of convergence is infinity. This means that the power series converges for any value of x.

Key Concepts

Understanding Power Series

A power series is a type of infinite series where each term is a power of a variable (often denoted as \(x\)). The general form of a power series is:

• \(\sum_{n=0}^{\infty} a_n x^n\), where \(a_n\) represents the coefficients of each term.

Each term of the series contains the variable \(x\) raised to a certain power, multiplied by the coefficient \(a_n\). Power series are extremely useful in mathematical analysis and can be used to represent functions in a way that is more convenient to work with.
The range of \(x\) values for which a power series converges is determined by its radius of convergence. This radius specifies the interval around a central point where the series behaves nicely and sums up to a finite value. Beyond this interval, the series may diverge, making it ineffective for calculations involving those \(x\) values.

Exploring the Ratio Test

The Ratio Test is a pivotal method in determining the convergence of series, particularly power series. To apply the Ratio Test, we consider the terms of the series denoted as \(a_n\). For a series to pass the Ratio Test:

• Calculate \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
• If the limit is less than 1, the series converges.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

This test works well for power series due to the nature of their terms, often involving factorials or exponential components which simplify nicely during the test. It's important to note that the Ratio Test helps find the radius of convergence in power series, informing us within what range of \(x\) the series can accurately represent a function.

Diving into Infinite Series

Infinite series extend the concept of a series into a sum with an endless number of terms. They are written in the form \(\sum_{n=0}^{\infty} a_n\), which indicates that terms keep adding infinitely.
Infinite series can either converge to a finite value or diverge depending on their terms’ nature.
Convergence is crucial as it implies that the series approaches a particular value as more terms are added. For power series, convergence may occur across certain intervals, defined by the radius of convergence.
Understanding infinite series can be enhanced by testing methods like the Ratio Test, which specifically help in ascertaining convergence properties. Finite sums are straightforward, but as we deal with infinitely many terms, these tests become essential to determine the behaviour of the series and ensure the series is useful for practical calculations.