Proof of simple formulae

Proof of simple formulae

Proof of Simple Formulae

Fundamental Techniques

• Familiarise yourself with induction, which is often used in proofs of simple formulae related to sequences and series.
• Remember that the principle of mathematical induction consists of two steps - base case, and inductive step.
• The base case is typically for the smallest value of ‘n’, which is usually 1.
• The inductive step assumes the formula holds true for ‘n=k’, and proves that it then must hold true for ‘n=k+1’.

Binomial Theorem

• Revisit the binomial theorem, which states that (x+y)^n expands into a series where the coefficients can be found using Pascal’s triangle or calculated as “n choose r”.
• Note that “n choose r” is equivalent to nCr which equals n! / r!(n-r)!.

Summation of Series Formulae

• Review the standard formula for the sum of an arithmetic series, Sn = n/2(2a + (n-1)d).
• Understand the formula for the sum of a finite geometric series, Sn = a(r^n - 1)/(r - 1) if r>1 or Sn = a(1 - r^n)/(1 - r) if r<1.
• Recall that the sum of squares and cubes can be derived using induction and binomial theorem.

Application in Generating Functions

• Understand that generating functions count the number of ways a certain task can be performed, simplifying problems with a complicated structure.
• Remember that generating functions provide a bridge between discrete and continuous mathematics and make it easier to prove simple formulae of series.
• Learn to encode sequences into power series to form the basis of the generating function.
• Solve problems related to these generating functions to find the sum of series.

Series Convergence Tests

• Revise key tests for determining convergence or divergence of series, including the ratio test, the root test, and the comparison test, which are essential to applying these formulae.
• Understand absolute convergence and conditional convergence.
• Be aware of the convergence of improper integrals in the context of these series formulae.