The Binomial Series: Extending the Validity?

The binomial series (1+x)n=1+nx+n(n−1)2!x2+... can be used to approximate √2\sqrt{2} by setting x=−1/2x=-1/2, giving the series 1−12n+14n(−1)n2+..., which converges, slowly.In summary, the binomial series (1+x)^n=1+nx+n(n-1)/2!x^2+... only converges for |x|<1 right? However, writing (1+x)^n differently (i.e. x^n(1+1/x)^n) can extend the validity of this series to include values of x such
  • #1
PFuser1232
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The binomial series ##(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + ...## only converges for ##|x| < 1## right?
Is it true that writing ##(1 + x)^n## differently (i.e. ##x^n (1 + \frac{1}{x})^n##) extends the validity of this series to include values of ##x## such that ##|x| > 1##?
 
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  • #2
MohammedRady97 said:
The binomial series (1+x)n=1+nx+n(n−1)2!x2+...(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + ... only converges for |x|<1|x| < 1 right?
No. A binomial series is only defined for a given n, Whatever the value of x, there exists an N such that [itex]nx>1000 [/itex] for n>N.
MohammedRady97 said:
Is it true that writing (1+x)n(1 + x)^n differently (i.e. xn(1+1x)nx^n (1 + \frac{1}{x})^n) extends the validity of this series to include values of xx such that |x|>1|x| > 1?
No.
 
  • #3
MohammedRady97 said:
The binomial series ##(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + ...## only converges for ##|x| < 1## right?

Yes. Although the series might converge for more values than just ##|x|<1##. Complete details are here: http://en.wikipedia.org/wiki/Binomial_series#Conditions_for_convergence

Is it true that writing ##(1 + x)^n## differently (i.e. ##x^n (1 + \frac{1}{x})^n##) extends the validity of this series to include values of ##x## such that ##|x| > 1##?

Yes, If ##|x|>1##, then your trick can be used. If ##|x|<1##, then your trick doesn't work, but the original series does of course. If ##|x|=1## then the situation is a bit annoying.
 
  • #4
Svein said:
No. A binomial series is only defined for a given n, Whatever the value of x, there exists an N such that [itex]nx>1000 [/itex] for n>N.

No.

micromass said:
Yes. Although the series might converge for more values than just ##|x|<1##. Complete details are here: http://en.wikipedia.org/wiki/Binomial_series#Conditions_for_convergence
Yes, If ##|x|>1##, then your trick can be used. If ##|x|<1##, then your trick doesn't work, but the original series does of course. If ##|x|=1## then the situation is a bit annoying.

Interesting. I once watched a video in which someone proves that the sum of all natural numbers is ##-\frac{1}{12}##, and in one of the steps, he substituted ##x = 1## into the binomial series. Apparently, anything is possible if you're Euler!
 
  • #5
MohammedRady97 said:
Interesting. I once watched a video in which someone proves that the sum of all natural numbers is ##-\frac{1}{12}##, and in one of the steps, he substituted ##x = 1## into the binomial series. Apparently, anything is possible if you're Euler!

Yes, that is valid, but only under conditions. For the standard convergence of series, plugging in ##x=1## is forbidden. But there are other definitions where series do not converge how we they usually do. Under those definitions, you do get ##-1/12##.
 
  • #6
micromass said:
Yes, that is valid, but only under conditions. For the standard convergence of series, plugging in ##x=1## is forbidden. But there are other definitions where series do not converge how we they usually do. Under those definitions, you do get ##-1/12##.

Is it possible to approximate ##\sqrt{2}## using the binomial series?
 
  • #7
Yes, but not directly. What you can do (for example) is to use ##x=-1/2## to approximate ##1/\sqrt{2} = \frac{\sqrt{2}}{2}##. And then you can easily find ##\sqrt{2}##.

Edit: I guess you can even do it directly, but convergence won't be very rapid.
 
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What is the binomial series and how is it used in mathematics?

The binomial series is a mathematical concept that allows for the expansion of a binomial expression (an expression with two terms) to an infinite series. It is used to approximate values of functions and solve mathematical problems involving binomial expressions.

What is the general formula for the binomial series?

The general formula for the binomial series is (x+y)^n = x^n + nx^(n-1)y + (n(n-1)/2!)x^(n-2)y^2 + (n(n-1)(n-2)/3!)x^(n-3)y^3 + ... + (n!/n!)y^n, where n is a positive integer and x and y are real numbers.

What is the range of values for which the binomial series is valid?

The binomial series is valid for all real values of x and y, as long as the value of x is between -1 and 1. This means that the series may not converge or give accurate results for values outside of this range.

How is the binomial series extended to include values outside of the valid range?

To extend the validity of the binomial series, the concept of analytic continuation is used. This involves using complex analysis techniques to expand the range of values for which the series is valid. By introducing imaginary numbers, the series can be extended to include values outside of the valid range.

What are some real-world applications of the binomial series?

The binomial series has many practical applications, including in physics, engineering, and finance. It is used to approximate values in complex physical systems, such as in quantum mechanics and thermodynamics. In finance, it is used to calculate compound interest and in engineering, it is used to model and analyze systems with multiple variables.

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