Method of Differences and Summation Techniques

Method of Differences and Summation Techniques

This lesson covers the following key concepts:

• Method of differences for series summation
• Telescoping series
• Summation using sigma notation
• Partial fractions method
• Series with general term involving factorials
• Series with terms in AP × GP pattern
• Advanced JEE-level summation problems

Important Formulas

• $$\sum_{k=1}^{n} [f(k) - f(k-1)] = f(n) - f(0) \text{ (telescoping)}$$
• $$\sum_{k=1}^{n} \frac{1}{k(k+1)} = 1 - \frac{1}{n+1}$$
• $$\sum_{k=1}^{n} \frac{1}{k(k+1)(k+2)} = \frac{1}{4} - \frac{1}{2(n+1)(n+2)}$$
• $$\sum_{k=1}^{n} k \cdot r^{k-1} = \frac{1 - (n+1)r^n + nr^{n+1}}{(1-r)^2}$$
• $$\sum_{k=1}^{n} k^2 r^{k-1} \text{ (using differentiation of GP)}$$
• $$\frac{1}{n(n+r)} = \frac{1}{r}\left(\frac{1}{n} - \frac{1}{n+r}\right)$$