Binomial series (radius of convergence)

In summary, a binomial series is an infinite series used to represent a function involving powers of a variable. Its radius of convergence, denoted by R, is the distance from the center of the series to the point of convergence and can be determined using the ratio test or the root test. The radius of convergence is significant because it determines the validity of the series for different values of x. In some cases, the radius of convergence can be infinite, meaning the series will converge and accurately represent the function for all values of x. However, this is not always the case and depends on the specific function and series.
  • #1
Fernando Revilla
Gold Member
MHB
631
0
I quote a question from Yahoo! Answers

Is the radius of convergence for all binomial series exactly 1?

I have given a link to the topic there so the OP can see my response.
 
Mathematics news on Phys.org
  • #2
If $|x|<1$, the binomial series $\displaystyle\sum_{k=0}^{\infty} \; {\alpha \choose k} \; x^k $ converges absolutely to $(1+x)^{\alpha}$ for any $\alpha\in\mathbb{R}$, but not always the radius of convergence is $1$. For example, if $\alpha$ is a non-negative integer, then the series is finite and the radius of convergence is $+\infty$.
 

FAQ: Binomial series (radius of convergence)

What is a binomial series?

A binomial series is an infinite series that is used to represent a function, typically involving powers of a variable. It is written in the form of (x+a)^n, where x is the variable, a is a constant, and n is a positive integer.

What is the radius of convergence for a binomial series?

The radius of convergence for a binomial series is the distance from the center of the series (a) to the point at which the series converges. It is typically denoted by R and can be calculated using the ratio test or the root test.

How do you determine the radius of convergence for a binomial series?

The radius of convergence can be determined using the ratio test or the root test. For the ratio test, you take the limit as n approaches infinity of |a(n+1)/a(n)|, where a(n) is the nth term of the series. If the limit is less than 1, the series converges, and the radius of convergence is the value of x at which the limit is 1. For the root test, you take the limit as n approaches infinity of the nth root of |a(n)|, and the same rules apply.

What is the significance of the radius of convergence in a binomial series?

The radius of convergence is important because it tells you the values of x for which the series is valid. If x is within the radius of convergence, the series will converge and accurately represent the function. However, if x is outside the radius of convergence, the series will diverge and will not accurately represent the function.

Can the radius of convergence of a binomial series ever be infinite?

Yes, the radius of convergence can be infinite in some cases. This means that the series will converge for all values of x and will accurately represent the function for all values of x. However, this is not always the case, and the radius of convergence will depend on the specific function and series being considered.

Similar threads

I Infinite Series of Infinite Series
Replies
7
Views
2K
MHB Identifying if an experiment is a binomial experiment
Replies
1
Views
811
MHB Theodore K's question at Yahoo Answers (Radius of convergence)
Replies
1
Views
2K
MHB Jenny's question at Yahoo Answers (Convergence of a series)
Replies
1
Views
1K
I Extend Euler Product Convergence Over Primes: Basic Qs
Replies
1
Views
1K
I The “philosophical cornerstone” of the Moller-Plesset perturbation theory
Replies
1
Views
971
I Complex Analysis Radius of Convergence.
Replies
7
Views
2K
MHB The Royal Primate's question at Yahoo Answers regarding convergence of a series
Replies
1
Views
2K
I Convergence and divergence of series and sequences
Replies
7
Views
535
MHB Emil's question at Yahoo Answers (Radius of convergence)
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top