Binominal Expansions as Infinite Sums

Binominal Expansions as Infinite Sums

Basic Concepts

• Binominal expansions as infinite sums involve converting a binominal expansion into an infinite series and perform analysis on it.

• The concept is based on the Binomial Theorem and holds for all values of

  x

  < 1.

The General Form

• The general form of a binominal expansion as an infinite sum is (1+x)^n = 1 + nx + n(n-1)x^2/2! + n(n-1)(n-2)x^3/3! + ….

Convergence

• This expansion is convergent for -1 < x < 1. This is critical when you assess the validity of your series.

• If you come across entries with

  x

  ≥ 1, it may cause the infinite sum to diverge or not exist.

Application to Calculus

Remember: Differentiating or integrating term by term gives new series that also converge for -1 < x < 1.

• Example: If (1+x)^n = 1 + nx + n(n-1)x^2/2! …, then d/dx (1+x)^n = n - n(n-1)x/1! + n(n-1)(n-2)x^2/2! ….
• Equally, a binomial series can be integrated term by term in the interval of convergence.

Dealing with Negative Powers

• The binomial series can be used with negative integral and fractional powers, not just positive integer powers.
• Example: (1+x)^(-n) = 1 - nx + n(n+1)x^2/2! - n(n+1)(n+2)x^3/3! + ….

Truncating the Series

• For

  x

  < 1, the terms in a binomial series become progressively smaller.

• This permits the truncation of the series after a few terms to provide an approximate equality if x is sufficiently small.