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Next: Bounding Summations Up: CSE 2320: Algorithms and Previous: Summations

Useful Series

Arithmetic Series


\begin{eqnarray*}\sum_{k=1}^n k & = & 1 + 2 + \ldots + n \\
& = & \frac{n(n+1)}{2} \\
& = & \Theta(n^2)
\end{eqnarray*}


Geometric (Exponential) Series


\begin{eqnarray*}\sum_{k=0}^n x^k & = & 1 + x + x^2 + \ldots + x^n \\
& = & \frac{x^{n+1} - 1}{x - 1}
\end{eqnarray*}



\begin{displaymath}\sum_{k=0}^{\infty} x^k = \frac{1}{1-x} \;\; {\rm if} \; \vert x\vert < 1 \end{displaymath}

Differentiating Series


\begin{displaymath}\frac{d}{dx} \sum_{k=0}^{\infty} x^k = \frac{d}{dx} \frac{1}{1-x} \end{displaymath}


\begin{displaymath}\sum_{k=0}^{\infty} kx^{k-1} = \frac{1}{(1-x)^2} \end{displaymath}


\begin{displaymath}x \sum_{k=0}^{\infty} kx^{k-1} = \sum_{k=0}^{\infty} kx^k =
\frac{x}{(1-x)^2} \end{displaymath}

Products


\begin{eqnarray*}\Pi_{k=1}^n a_k & = & a_1 * a_2 * \ldots * a_n \\
& = & 1 \; {\rm if} \; n = 0
\end{eqnarray*}



\begin{displaymath}\lg \left( \Pi_{k=1}^n a_k \right) = \sum_{k=1}^n \lg a_k \end{displaymath}



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