Question

How do you find the radius of convergence?

Answer 1

The solution will be done in 3 different steps.
Considering ${a_{n + 1}} = {c_{n + 1}} \times {\left( {x - a} \right)^{n + 1}}$ and ${a_{n + 1}} = {c_{n + 1}} \times {\left( {x - a} \right)^{n + 1}}$. Now, we will divide these two to get the ratio and further simplify. After that we will individually calculate the radius of convergence for 3 different cases:-
1. If Limit=0
2. If limit$= N \times |x - a|$
3. Limit= $\infty$

Complete step by step solution:
Here, in order to find the radius of convergence, we will use a ratio test.
In this method our steps to follow will be:
Step 1:
Consider ${a_{n + 1}} = {c_{n + 1}} \times {\left( {x - a} \right)^{n + 1}}..........(1)$
Replacing n by n+1, we will get
${a_{n + 1}} = {c_{n + 1}} \times {\left( {x - a} \right)^{n + 1}}..........(2)$
Step 2:
We will now divide both (1) and (2) to simplify the ratio further. This will give us:-
$\dfrac{{{a_{n + 1}}}}{{{a_n}}} = \dfrac{{{c_{n + 1}} \times {{\left( {x - a} \right)}^{n + 1}}}}{{{c_n} \times {{\left( {x - a} \right)}^n}}} \\\ = \dfrac{{{c_{n + 1}}}}{{{c_n}}} \times \left( {x - a} \right) \\\$
Step 3:
Calculate the limit of the above ratio. Considering $n \to \infty$
Step 4:
Calculate the result according to below mentioned rules for:-
Case 1:
If Limit=0 of the absolute value of ratio if $n \to \infty$
Radius of convergence- $\lim it = N \times |x - a|$
Power series will be converged for every possible value of x.
Case 2:
If limit$= N \times |x - a|$where N=any finite but positive number
Radius of convergence=$\dfrac{1}{N}$. Whereas the interval for the convergence will be $\left( {a - \dfrac{1}{N},a + \dfrac{1}{N}} \right)$.
Case 3:
Limit=$\infty$
Radius of convergence will be equal to 0. The power series will converge at x=a. And this will be the sole where it will converge.

Note: Always remember that the interval may or may not include the extreme points. In order for you to confirm the above fact, you will have to put the values in the power series and expand it to form the infinite series. After that, perform the convergence test to confirm the given question.