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