Does the exponent in a series affect its radius of convergence?

In summary, the discussion is about finding the radius of convergence of a series with a power of 3 in the exponent. The confusion arises because the index of summation is not specified. The problem can be solved using the ratio test, which results in a radius of 3. However, if the power was just z^n, the radius of convergence would be 27.
  • #1
ModernLogic
Hi folks. I need to find the radius of convergence of this series: [tex]\sum_{k=0}^\infty \frac{(n!)^3z^{3n}}{(3n)!} [/tex]

The thing throwing me off is the [tex]z^{3n}[/tex]. If the series was [tex]\sum_{k=0}^\infty \frac{(n!)^3z^n}{(3n)!} [/tex] I can show it has radius of convergence of zero. But [tex]z^{3n}[/tex] means its only taking power multiples of 3. Does that change anything?

Thanks.
 
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  • #2
is the index of summation k or n?
 
  • #3
The problem can still be solved using the ratio test:
[tex]\lim_{n\rightarrow\infty}\frac{((n+1)!)^3z^{3n+3}}{(3n+3)!}\times\frac{(n!)^3}{z^{3n}(3n)!}[/tex]
=[tex]\lim_{n\rightarrow\infty}\frac{(n+1)^3z^{3}}{(3n+3)(3n+2)(3n+1)}[/tex]
Now, being sloppy so that I don't have to write so much, in the limit this is going to be equal to:
[tex]\lim_{n\rightarrow\infty}\frac{(n)^3z^{3}}{27(n)^3}[/tex]
=[tex]\lim_{n\rightarrow\infty}\frac{z^{3}}{27}[/tex]
=[tex]\frac{z^{3}}{27}[/tex]
so, requiring the absolute value of this expression to be less than 1 gives a radius of 3
 
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  • #4
I got the exact same result, Leonhard, throught D' Alambert's Criterium (ratio test), but given he said a radius of convergence of 0 for z^n, it confused me to what is exactly the index of summation k or n.
 
  • #5
Yeah, I didn't even catch that, but its probably just a mistake.
 
  • #6
Modern Logic: The only difference between having z3n instead of zn in the problem is that you will have z3 instead of z in the final formula- Take the cube root. (Let y= z3 so that the sum involves yn.)

However, you are wrong when you say that if the problem involved zn instead of z3n you would get a radius of convergence of 0. As Leonard Euler said, you would get 27 and so for z, the cube root of that, 3.
 

What is the "Radius of Convergence"?

The radius of convergence is a mathematical concept used in power series to determine the values of the independent variable for which the series converges to a finite value.

How is the "Radius of Convergence" calculated?

The radius of convergence is calculated by using the ratio test, which compares the absolute value of the ratio of successive terms in the series to a certain threshold value.

What does a larger "Radius of Convergence" indicate?

A larger radius of convergence indicates that the power series has a wider range of values for which it converges, making it more useful for approximating functions over a larger domain.

What happens if the "Radius of Convergence" is infinite?

If the radius of convergence is infinite, the power series converges for all values of the independent variable, making it a useful tool for approximating functions over their entire domain.

Can the "Radius of Convergence" change for different power series?

Yes, the radius of convergence can change for different power series, as it is dependent on the coefficients and the independent variable in the series.

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