Explain binomial series

The Binomial Theorem is the method of expanding an expression which has been raised to any finite power. A binomial Theorem is a powerful tool of expansion, which has application in Algebra, probability, etc.

Binomial expansion

The Binomial Expansion Theorem is an algebra formula that describes the algebraic expansion of powers of a binomial. According to the binomial expansion theorem, it is possible to expand any power of x + y into a sum of the terms. The Binomial Expansion Formula or Binomial Theorem is given as:

[large (x+y)^{n} = x^{n} + nx^{n-1}y + frac{n(n-1)}{2!} x^{n-2} y^{2} + … + y^{n}]

Few other binomial expansions

  • (x + y)n + (x−y)n = 2[C0 xn + C2 xn-1 y2 + C4 xn-4 y4 + …]
  • (x + y)n – (x−y)n = 2[C1 xn-1 y + C3 xn-3 y3 + C5 xn-5 y5 + …]
  • (1 + x)n = nΣr-0 nCr . xr = [C0 + C1 x + C2 x2 + … Cn xn]
  • (1+x)n + (1 − x)n = 2[C0 + C2 x2+C4 x4 + …]
  • (1+x)n − (1−x)n = 2[C1 x + C3 x3 + C5 x5 + …]
  • The number of terms in the expansion of (x + a)n + (x−a)n are (n+2)/2 if “n” is even or (n+1)/2 if “n” is odd.
  • The number of terms in the expansion of (x + a)n − (x−a)n are (n/2) if “n” is even or (n+1)/2 if “n” is odd.

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