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\section{Solving the sums, mathematical correct estimate}  \subsection{Laws and forumulas of summation}  These summation laws/formulas are used frequently.  They are very common and thus we won't prove any of them.  Most of them can be found \href{http://en.wikipedia.org/wiki/Summation}{here}  \begin{enumerate}  \item $\displaystyle\sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n)$  \item $\displaystyle\sum_{n=s}^t f(n) + \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) + g(n)\right]$  \item $\displaystyle\sum_{n=s}^t f(n) - \sum_{n=s}^{t} g(n) = \sum_{n=s}^t \left[f(n) - g(n)\right]$  \item $\displaystyle\sum_{i=m}^n 1 = n+1-m \Rightarrow \sum_{i=m}^{n-1} 1 = n-m$  \item $\displaystyle\sum_{i=m}^n i = \frac{n(n+1)}{2} - \frac{m(m-1)}{2} \Rightarrow \sum_{i=m}^{n-1} i = \frac{n(n-1)}{2} - \frac{m(m-1)}{2}$  \item $\displaystyle\sum_{i=0}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}$  \end{enumerate}  \begin{equation}  \begin{split}  m &= \sqrt{n}\\