A summation is the mathematical process of adding a sequence of numbers. It is often represented using the sigma notation (\(\Sigma\)), which provides a compact way to express long series. Summations are foundational in algebra, calculus, number theory, and statistics.
\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]
\[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]
\[ \sum_{k=1}^{n} k^3 = \left( \frac{n(n+1)}{2} \right)^2 \]
\[ \sum_{k=1}^{n} k^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30} \]
\[ \sum_{k=1}^{n} (2k - 1) = n^2 \]
\[ \sum_{k=1}^{n} 2k = n(n+1) \]
\[ \sum_{k=p}^{q} k = \frac{(q + p)(q - p + 1)}{2} \]