# Special Power Series

## 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

$$\frac{1}{1-x} = 1 + x + x^2 +x^3 + \cdots \quad(\text{for } -1 < x < 1)$$

$$\frac{1}{1+x} = 1 – x + x^2 – x^3 + \cdots \quad(\text{for } -1 < x < 1)$$

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

$$\ln(1+x) = x – \frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \cdots \quad (\text{for } -1 < x < 1)$$

$$\sin x = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots$$

$$\cos x = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots$$

$$\tan\,x = x – \frac{x^3}{3} + \frac{2x^5}{15} – \frac{17x^7}{315} + \cdots \quad \left(\text{for } -\frac{\pi}{2} < x < \frac{\pi}{2} \right)$$

$$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots$$

$$\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots$$