Sum of Series

Memorizing formulas for the sums of special series is crucial for solving many problems quickly. Here are some of the most common ones:

• First m Integers: $\sum_{n=1}^{m} n = \frac{m(m+1)}{2}$
• First m Odd Integers: $\sum_{n=1}^{m} (2n-1) = m^2$
• First m Even Integers: $\sum_{n=1}^{m} 2n = m(m+1)$
• First m Squares: $\sum_{n=1}^{m} n^2 = \frac{m(m+1)(2m+1)}{6}$
• First m Cubes: $\sum_{n=1}^{m} n^3 = (\frac{m(m+1)}{2})^2$
• Infinite Geometric Series: $\sum_{n=0}^{\infty} a_1 d^n = \frac{a_1}{1-d}$ for $|d| < 1$