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