Derive PI -Using the Binomial Theorem

Let’s get prepared and party for the forthcoming π Day!

Start With Pascal’s Triangle

Each entry in Pascal’s Triangle is the number of combinations of k from n elements, and that is precisely the binomial coefficient C(n, k). It is also the number of combinations of powers of x partitioning in the binomial expansion.
Pascal’s Triangle up to 6 rows is:

          1
        1   1
      1   2   1
    1   3   3   1
  1   4   6   4   1
1   5  10  10   5   1

• Row n will haven+1 numbers, which are the combination numbers C(n, k).
• These combination numbers satisfy the recurrence relation:

C(n, k) = C(n-1, k-1) + C(n-1, k)

• The number C(n, k) is the entry in the n-th row, k-th column of Pascal’s Triangle.
• This is a finite sum with only n+1 terms.

Binomial Theorem

For non-negative integer n:

(1 + x)^n = ∑[k=0 to n] C(n,k)x^k

Each term of the sum is the number of ways of picking k terms of x when the multiplicative expansion of (1 + x)^n is performed.

where:

C(n,k) = n!/(k!(n-k)!)

Isaac Newton first obtained the generalized binomial theorem, which allows α to be any real number, producing an infinite series expansion.