Radius of Convergence for Series: 3n+3 vs 3n+3!

  • Thread starter jack.o
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In summary, the conversation discusses the equation (3n+3)!=(3n)!+3! and its relation to finding the radius of convergence for a series. The equation is simplified and the concept of using the ratio test to determine convergence is mentioned.
  • #1
jack.o
3
0
Is (3n+3)!=(3n)!+3!
? probably obvious but I'm not certain. Trying to work out a radius of convergence for a series.
 
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  • #2
jack.o said:
Is (3n+3)!=(3n)!+3!
? probably obvious but I'm not certain. Trying to work out a radius of convergence for a series.

No. It is the product from i=1 to 3 of (3n!)*(3n+i)
 
  • #3
You can quickly verify this for yourself by checking the case where n = 1.
 
  • #4
(3n+3)!=(3n+3)(3n+2)(3n+1)(3n)(3n-1)...(6)(5)(4)(3)(2)(1)
 
  • #5
Ok, this convergence question still has me stuck

[tex]\stackrel{\infty}{\stackrel{\sum}{n=0}}[/tex][tex]\stackrel{\chi^{n}}{\overline{(3n)!}}[/tex]

Got the n+1 term and tried dividing the nth term by the nth+1 but does not seem to cancel nicely.
 
Last edited:
  • #6
If [tex]\stackrel{\chi^{n}}{(3n)!}[/tex] is not a fraction with the line missing, then I have no idea what you mean.
 
  • #7
It is meant to be a fraction, not used to the equation editor software here.
 
  • #8
Jack - you were on the right track by figuring out the n+1 term. Use the ratio test. You should see fairly readily that the series converges to 0 as n goes to infinity.
 

What is the definition of the radius of convergence for a series?

The radius of convergence for a series is a value r such that the series converges for all values of x within a distance of r from the center of the series, and diverges for all values of x outside of this distance.

How is the radius of convergence calculated for a series?

The radius of convergence is typically found by applying the ratio test, in which the limit of the absolute value of the terms of the series is taken as n approaches infinity. If this limit is less than 1, the series converges, and the radius of convergence can be calculated as the reciprocal of the limit. If the limit is greater than 1 or infinite, the series diverges and the radius of convergence is 0.

What is the significance of the radius of convergence for a series?

The radius of convergence is an important value for determining the behavior of a series. If the value of x falls within the radius of convergence, the series will converge and the terms will approach a finite value. If x lies outside of the radius of convergence, the series will diverge and the terms will approach infinity. The radius of convergence also helps to determine the interval of convergence for a series.

How does the series 3n+3 compare to the series 3n+3! in terms of their radii of convergence?

The series 3n+3 and 3n+3! have the same radius of convergence, which is equal to 1. This means that both series will converge for all values of x within a distance of 1 from the center of the series and diverge for all values of x outside of this distance. However, the intervals of convergence may differ for these two series.

What happens if the radius of convergence for a series is equal to 0?

If the radius of convergence for a series is equal to 0, then the series will converge only at the center point and will diverge for all other values of x. This is because the radius of convergence represents the distance from the center of the series within which the series will converge. If the radius is 0, then the series will only converge at the center point and diverge everywhere else.

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