Find the radius of convergence

In summary, the radius of convergence is a mathematical concept used in power series to determine the range of values for which the series will converge. It is calculated using the ratio test and is important for determining the validity of a power series and making predictions or calculations based on it. When the radius of convergence is infinite, the series will converge for all values of the independent variable. However, the radius of convergence can change depending on the function being represented and can be affected by operations such as differentiation or integration. It is always important to check the radius of convergence when performing such operations on a power series.
  • #1
JulioMarcos
2
0
Homework Statement
Find the radius of convergence of the Series:
[tex]\sum_{i=1}^{\infty}\frac{(2n)!x^n}{(n!)^2}[/tex]

The attempt at a solution
I used the Ratio Test but I always get L = [tex]|\frac{2x}{n+1}|[/tex]

The answer is 1/4. I think I am mistaking with factorial.
 
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  • #2
Solved. The problem was tha
(2(n+1))! = (2n + 2)(2n + 1)(2n)!
and I was doing (2(n+1))! = (2n + 2)(2n)!
because i thought you should distribute the 2
 
  • #3
Ah it is always good to find your own mistake. Off the bat that was my guess... Sadly I got in the thread too late.
 

What is the radius of convergence?

The radius of convergence is a mathematical concept used in power series to determine the range of values for which the series will converge. It is represented by the letter "R" and is typically given in terms of a positive real number.

How is the radius of convergence calculated?

The radius of convergence is calculated using the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms in the power series. If this limit is less than 1, then the series will converge, and the radius of convergence can be determined from the resulting equation.

Why is the radius of convergence important?

The radius of convergence is important because it tells us which values of the independent variable will result in a convergent series. This is crucial for determining the validity of a power series and for making predictions or calculations based on the series.

What happens when the radius of convergence is infinite?

If the radius of convergence is infinite, it means that the power series will converge for all values of the independent variable. This is the ideal scenario, as it allows us to make predictions and perform calculations for any input value without worrying about the series diverging.

Can the radius of convergence change?

Yes, the radius of convergence can change depending on the function being represented by the power series. It can also be affected by operations such as differentiation or integration. It is important to check the radius of convergence when performing such operations on a power series.

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