Binomial Series Simplification

In summary, a binomial series is a mathematical expression that represents the expansion of a binomial raised to a positive integer power. The coefficients in a binomial series can be found using the binomial theorem formula. Its main purpose is to simplify and express complicated algebraic expressions. It is different from a Taylor series as it only involves binomial functions. A binomial series can also be used to find the value of irrational numbers, as long as they can be expressed as a binomial function.
  • #1
courtrigrad
1,236
2
How does one deduce the following:

We are given

[tex] \binom{-3}{n} = \frac{(-3)(-4)(-5)\cdot\cdot\cdot\cdot\cdot(-3-n+1)}{n!} = \frac{(-3)(-4)(-5)\cdot\cdot\cdot\cdot[-(n+2)]}{n!} [/tex]. How do we get from here to:
[tex] \frac{(-1)^{n}\cdot2\cdot3\cdot4\cdot5\cdot\cdot\cdot\cdot(n+1)(n+2)}{2\cdot n!} = \frac{(-1)^{n}(n+1)(n+2)}{2} [/tex]?Thanks
 
Last edited:
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  • #2
nevermind. I figured it out
 

FAQ: Binomial Series Simplification

What is a binomial series?

A binomial series is a mathematical expression that represents the expansion of a binomial (a polynomial with two terms) raised to a positive integer power. It is also known as a binomial theorem or binomial expansion.

How do you find the coefficients in a binomial series?

The coefficients in a binomial series can be found by using the binomial theorem formula: (x + y)^n = nC0x^n + nC1x^(n-1)y + nC2x^(n-2)y^2 + ... + nCnx^0y^n, where n is the power and nCr is the binomial coefficient (also known as the combination).

What is the purpose of using a binomial series?

The main purpose of using a binomial series is to simplify and express complicated algebraic expressions in a more manageable and organized form. It is also used in various fields of mathematics, such as calculus and statistics, to solve problems and make predictions.

How is a binomial series different from a Taylor series?

A binomial series is a special case of a Taylor series, where the function being expanded is a binomial function. In a Taylor series, the function can be any polynomial or transcendental function, while a binomial series only involves binomial functions.

Can a binomial series be used to find the value of irrational numbers?

Yes, a binomial series can be used to find the value of irrational numbers, as long as the irrational number can be expressed as a binomial function. For example, the square root of 2 can be approximated using a binomial series expansion of (1+x)^1/2, where x is a small value close to 0.

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