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# Math formulas:Special power series

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### Powers of Natural Numbers

 $$\sum\limits_{k=1}^n k = \frac{1}{2}n(n+1)$$
 $$\sum\limits_{k=1}^n k^2 = \frac{1}{6}n(n+1)(2n+1)$$
 $$\sum\limits_{k=1}^n k^3 = \frac{1}{4}n^2(n+1)^2$$

### Special Power Series

 $$\frac{1}{1-x} = 1 + x + x^2 +x^3 + \cdots \quad(\text{for } -1 < x < 1)$$
 $$\frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots \quad(\text{for } -1 < x < 1)$$
 $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$
 $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad (\text{for } -1 < x < 1)$$
 $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
 $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$
 $$\tan\,x = x - \frac{x^3}{3} + \frac{2x^5}{15} - \frac{17x^7}{315} + \cdots \quad \left(\text{for } -\frac{\pi}{2} < x < \frac{\pi}{2} \right)$$
 $$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots$$
 $$\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots$$