B Useful Series Results

B.1 \(e^x\)

In many calculus books, the following is shown (for any \(x\)): \[e^x = \sum_{i=0}^\infty \frac{x^i}{i!}\]

Another series result for \(e^x\) is \[e^x = \lim_{n \to \infty} \left( 1 + \frac{x_n}{n} \right)^n \;\;\;\;\; \textrm{ if } \;\; x_n \to x\]

B.2 Geometric Series

The geometric series result is that \[\sum_{x=0}^\infty \alpha^x = \frac{1}{1-\alpha}\;\;\;\; \textrm{ if } \;\;\;\vert\alpha\vert<1\]

By repeatedly differentiating both sides we can derive \[\sum_{x=1}^\infty x \alpha ^{x-1} = \frac{1}{ (1-\alpha)^2 }\;\;\;\; \textrm{ if } \;\;\;\vert\alpha\vert<1\] and \[\sum_{x=2}^\infty x(x-1) \alpha^{x-2} = \frac{2}{(1-\alpha)^3}\;\;\;\; \textrm{ if } \;\;\;\vert\alpha\vert<1\]

If the series isn’t all the way to \(\infty\), the the following is handy result. \[\sum_{x=0}^{k-1} \alpha^x = \frac{1-\alpha^k}{1-\alpha}\;\;\;\; \textrm{ if } \;\;\; \alpha \ne 1\]