# Find Radius of Convergence from Recursion Equation

In summary, to find the radius of convergence for the power series solution of the given recursion equation, we can use the ratio test and simplify the ratio using the given recursion equation. This results in a radius of convergence of 0, meaning the power series solution will converge at the origin.
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## Homework Statement

Find Radius of Convergence of the corresponding power series solution from Recursion Equation alone:

$$n^2a_{n+2} - 3(n+2)a_{n+1} +3a_{n-1} = 0 \qquad(1)$$

## Homework Equations

R = 1/L where

$$L = \lim_{n\rightarrow\infty}\left|{\frac{a_{k+1}}{a_k}\right|\qquad(2)$$

## The Attempt at a Solution

I was thinking that I could solve (1) for an+1 and then solve it again for an and then use the ratio in (2). But I feel like that might be a very illegal move.

Thoughts on this?

To find the radius of convergence for the power series solution of the given recursion equation, you can use the ratio test as outlined in (2). However, instead of solving for an+1 and an separately, you can use the given recursion equation to simplify the ratio.

First, divide both sides of the recursion equation by an to get:

n^2a_{n+2}/a_n - 3(n+2)a_{n+1}/a_n + 3a_{n-1}/a_n = 0

Then, using the fact that L = lim_{n\rightarrow\infty}\left|{\frac{a_{k+1}}{a_k}\right|, we can rewrite the above equation as:

n^2(a_{n+2}/a_n) - 3(n+2)(a_{n+1}/a_n) + 3(a_{n-1}/a_n) = 0

Now, as n approaches infinity, the terms (a_{n+2}/a_n) and (a_{n-1}/a_n) will approach zero, leaving us with:

0 - 3L + 3(0) = 0

Solving for L, we get L = 0.

Therefore, the radius of convergence for the power series solution is R = 1/L = 1/0 = undefined. This means that the power series solution will converge at the point where the ratio of consecutive terms is equal to 0, which is at the origin.

I hope this helps! Let me know if you have any further questions.

Scientist

## What is the definition of convergence in mathematics?

Convergence is a mathematical concept that refers to the idea of a sequence or series of numbers approaching a specific value or limit as the number of terms increases.

## What is a recursion equation?

A recursion equation is a mathematical formula that defines a sequence or series of numbers by relating each term to one or more of the previous terms.

## What is the radius of convergence?

The radius of convergence is a measure of how quickly a sequence or series of numbers approaches its limit. It is the distance from the center of a power series to the nearest point where the series converges.

## How do you find the radius of convergence from a recursion equation?

To find the radius of convergence from a recursion equation, you can use the ratio test or the root test. These tests involve taking the limit of the ratio or root of consecutive terms in the sequence to determine if the series converges or diverges.

## Why is finding the radius of convergence important?

Finding the radius of convergence is important because it allows us to determine the range of values for which a sequence or series is convergent. This information is used in many areas of mathematics, including calculus, differential equations, and complex analysis.

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