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\[ 1 - 2 + 3 - \cdots + \left( { - 1} \right)^{n - 1} n = \left\{ {\begin{array}{*{20}c} {\frac{1}{2}\left( {n + 1} \right)n,n为奇数} \\ { - \frac{n}{2},n为偶数} \\ \end{array}} \right. \]
\[ 1^2 - 2^2 + 3^2 - \cdots + \left( { - 1} \right)^{n - 1} n^2 = \left( { - 1} \right)^{n - 1} \frac{1}{2}n\left( {n + 1} \right) \]
\[ 1^3 - 2^3 + 3^3 - \cdots + \left( { - 1} \right)^{n - 1} n^3= \left\{ {\begin{array}{*{20}c} {\frac{1}{4}\left( {2n - 1} \right)\left( {n + 1} \right)^2 ,n为奇数} \\ { - \frac{1}{4}n^2 \left( {2n + 3} \right),n为偶数} \\ \end{array}} \right. \]
\[ 1^4 - 2^4 + 3^4 - \cdots + \left( { - 1} \right)^{n - 1} n^4 = \left( { - 1} \right)^{n - 1} \frac{1}{2}n\left( {n + 1} \right)\left( {n^2 + n - 1} \right) \]