Problem 378 - Advanced Calculus

Question

Find the radius of convergence of the following power series: a) \(^{\infty} \sum_{n=1}\left(x^{n} / n\right)\) b) \({ }^{\infty} \Sigma_{\mathrm{n}=1}\left\\{(2 \mathrm{n}) ! /(\mathrm{n} !)^{2}\right\\} \mathrm{x}^{\mathrm{n}}\) c) \(^{\infty} \Sigma_{\mathrm{n}=1}\left\\{(3 \mathrm{n}) ! /(\mathrm{n} !)^{2}\right\\} \mathrm{x}^{\mathrm{n}}\)

Step-by-Step Solution

Answer

The radius of convergence for the given power series are: a) 1 b) 1/4 c) 1/27

Step 1: Find the ratio of consecutive terms

We will find the absolute value of the ratio of consecutive terms: \(|\frac{a_{n+1}}{a_n}| = \frac{|\frac{x^{n+1}}{n+1}|}{|\frac{x^n}{n}|}\)

Step 2: Simplify the ratio

After simplifying the given expression, we get: \(|\frac{a_{n+1}}{a_n}| = \frac{x^{n+1}n}{x^n(n+1)} = \frac{x^n * x * n}{x^n * (n+1)} = \frac{n}{n+1}|x|\)

Step 3: Apply the limit

Apply the limit as n approaches infinity: \(\lim_{n \to \infty} \frac{n}{n+1}|x| = |x|\)

Step 4: Determine the radius of convergence

According to the Ratio Test, the series converges when the limit of the ratio is less than 1. In this case, we have: \(|x| < 1\) The radius of convergence is 1. b) \(a_n = \frac{(2n)!}{(n!)^2} x^n\)

Step 1: Find the ratio of consecutive terms

We will find the absolute value of the ratio of consecutive terms: \(|\frac{a_{n+1}}{a_n}| = \frac{|\frac{(2(n+1))!}{((n+1)!)^2}x^{n+1}|}{|\frac{(2n)!}{(n!)^2}x^n|}\)

Step 2: Simplify the ratio

After simplifying the given expression, we get: \(|\frac{a_{n+1}}{a_n}| = \frac{(2(n+1))!x^{n+1}}{(2n)!(n+1)^2x^n} = \frac{(2n+2)!}{(2n)!(n+1)^2}|x|\)

Step 3: Apply the limit

Apply the limit as n approaches infinity: \(\lim_{n \to \infty} \frac{(2n+2)!}{(2n)!(n+1)^2}|x| = 4 |x|\)

Step 4: Determine the radius of convergence

According to the Ratio Test, the series converges when the limit of the ratio is less than 1. In this case, we have: \(4|x| < 1\) The radius of convergence is 1/4. c) \(a_n = \frac{(3n)!}{(n!)^2} x^n\)

Step 1: Find the ratio of consecutive terms

We will find the absolute value of the ratio of consecutive terms: \(|\frac{a_{n+1}}{a_n}| = \frac{|\frac{(3(n+1))!}{((n+1)!)^2}x^{n+1}|}{|\frac{(3n)!}{(n!)^2}x^n|}\)

Step 2: Simplify the ratio

After simplifying the given expression, we get: \(|\frac{a_{n+1}}{a_n}| = \frac{(3(n+1))!x^{n+1}}{(3n)!(n+1)^2x^n} = \frac{(3n+3)!}{(3n)!(n+1)^2}|x|\)

Step 3: Apply the limit

Apply the limit as n approaches infinity: \(\lim_{n \to \infty} \frac{(3n+3)!}{(3n)!(n+1)^2}|x| = 27 |x|\)

Step 4: Determine the radius of convergence

According to the Ratio Test, the series converges when the limit of the ratio is less than 1. In this case, we have: \(27|x| < 1\) The radius of convergence is 1/27.

Key Concepts

Power Series

A power series is like a polynomial that goes on forever. Imagine playing with building blocks, where each block is a term like \(a_n x^n\), and you keep stacking them indefinitely to create a very long tower. In mathematics, we write this unending tower as \( \sum_{n=0}^\infty a_n x^n \), where each \(a_n\) is a coefficient of the series, \(x\) is a variable, and \(n\) is the exponent that increases by 1 with each additional term. The power series can converge (which means it has a finite sum) or diverge (it goes off to infinity) within certain values of \(x\). The range of \(x\) values where the series converges is very important and is determined by the radius of convergence. Think of it as the safe zone where your tower won't topple over.

Understanding power series is crucial when dealing with functions in calculus, as power series can represent functions in a form that is easier to manipulate and analyze. Students should practice expanding functions into power series and determine the interval where these series are valid representations of the function.

Ratio Test

The ratio test is like a tool we use to figure out whether our endlessly tall tower of blocks (the power series) is stable or not. Specifically, we look at the size of one block compared to the one immediately below it as the tower grows taller, and this analogy represents the ratio of consecutive terms in a series. To apply the ratio test, we calculate the limit of the absolute value of the ratio of the term \(a_{n+1}\) to the term \(a_n\) as \(n\) goes to infinity, denoted as \( \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| \).

If the limit is less than 1, our tower stands firm and the series converges. If it's exactly 1, the test is inconclusive—it's like not being sure whether the tower will stand or fall. And if the limit is greater than 1, the tower—and therefore the series—falls, and we say it diverges. Applying the ratio test helps you avoid the mess of a falling tower by letting you determine the stability (convergence) of the series beforehand.

Limit of a Sequence

Imagine you're walking closer and closer to a wall with each step. The wall is the limit you're approaching, and your steps are the terms in a sequence. Mathematically, we express this approach with \( \lim_{n \to \infty} a_n \), which reads 'the limit of the sequence \(a_n\) as \(n\) approaches infinity.' The idea is to see what value the terms of the sequence get closer to as \(n\) becomes very large.

It's a bit like trying to find out how close you can get to the wall without actually touching it. If your steps (terms in the sequence) stay consistent and predictable, you have a clear limit. However, if your approach becomes erratic or doesn't settle down (like steps varying wildly in size), you might not have a limit at all. Limits are fundamental in calculus because they help define derivatives and integrals, the pillars on which calculus stands.

Factorial Notation

Factorial notation is a convenient way to say 'multiply all integers from 1 to \(n\) together.' We use an exclamation point to denote this: \( n! \), known as 'n factorial.' So, if \( n = 5 \), then \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \). It's like saying you want to take 5 steps, and you want to know how many different ways you could mix up those steps.

Factorials grow super fast. This means that even for relatively small numbers, their factorials are huge. They're particularly important when dealing with power series that have factorial terms because they have a big influence on the series' convergence. For example, in combinatorics, factorials are used to count how many ways you can arrange things, and in calculus, they're often found in series expansions of functions, like the Taylor or Maclaurin series.