Series

• Geometric Series (finite): $\sum_{k=0}^{n-1} ar^k=a + ar + ar^2 + \cdots + ar^{n-1}=a\frac{1-r^n}{1-r}\qquad r \neq 1$
• Geometric Series: $\sum_{n=0}^{\infty} ar^n=a + ar + ar^2 + ar^3 + \cdots=\frac{a}{1-r}\qquad |r| < 1$
• Binomial Theorem: $(1+x)^n=\sum_{k=0}^{\infty}\binom{n}{k}x^k=1 + nx+ \frac{n(n-1)}{2!}x^2+ \frac{n(n-1)(n-2)}{3!}x^3+ \cdots$
• Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$