Math Series Results

1. 1 + 2 + 3 +\cdots + n = \frac{{n(n + 1)}}{2}

2. {1^2} + {2^2} + {3^2} + \cdots  + {n^2} = \frac{{n(n + 1)(2n + 1)}}{6}

3. {1^3} + {2^3} + {3^3} + \cdots  + {n^3} = \frac{{{n^2}{{(n + 1)}^2}}}{4}

4. {1^4} + {2^4} + {3^4} + \cdots  + {n^4} = \frac{{n(n + 1)(2n + 1)(3{n^2} + 3n - 1)}}{{30}}

5. 2 + 4 + 6 +\cdots + 2n = n(n + 1)

6. 1 + 3 + 5 +\cdots + (2n - 1) = {n^2}

7. 1 + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \cdots  = \frac{{{\pi ^2}}}{6} = 1.64493

8. 1 + \frac{1}{{{2^3}}} + \frac{1}{{{3^3}}} + \cdots  = 1.20205

9. 1 + \frac{1}{{{2^4}}} + \frac{1}{{{3^4}}} + \cdots  = \frac{{{\pi ^4}}}{{90}} = 1.08232

10. 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots  = {\log _e}2 = 0.6931

11. 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi }{4}

12. 1 - \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} -  \frac{1}{{{4^2}}} + \cdots = {\pi ^2}

13. 1 + \frac{1}{{{3^2}}} + \frac{1}{{{5^2}}} + \cdots  = \frac{{{\pi ^2}}}{8}

14. 1 + 1 + \frac{1}{{2!}} + \frac{1}{{3!}} + \cdots  = e