Bored of solving quadratic equations? Can't be bothered with cubics? Then it's time to step into the infinite — and marvel at the fact that many of the functions you'll have come across can be expressed using infinite sums made of powers of $x.$ A great example are the trigonometric functions sine and cosine. It turns out that they can be expressed as follows $$\cos{(x)} = 1 - \frac{x^2}{2!} + \frac{x^4}{4 !} - \frac{x^6}{6!} + ... $$ $$\sin{(x)} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... ,$$ where $n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1.$ For both of these series the beautiful pattern continues indefinitely. Choosing a particular value for $x,$ you will find that the infinite series converges to $\cos{(x)}$ and $\sin{(x)}$ respectively. Similarly beautiful series can be used to express the exponential function $e^x$ and the natural logarithm $ln(x)$: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3! } + \frac{x^4}{4!} + \frac{x^5}{5! } + ...$$ and $$\ln{(x)} = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} + \frac{(x-1)^5}{5} - ...,$$ though in the case of the logarithm this only works when $0

The black curve is the graph of $\cos{(x)}.$ The cyan curve is the graph of $$f(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4 !} - \frac{x^6}{6!} + \frac{x^8}{8!}. $$ The purple curve is the graph of $$f(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4 !} - \frac{x^6}{6!} + \frac{x^8}{8!}- \frac{x^{10}}{10!}+ \frac{x^{12}}{12!}. $$ And the red curve is the graph of $$f(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4 !} - \frac{x^6}{6!} + \frac{x^8}{8!}- \frac{x^{10}}{10!}+ \frac{x^{12}}{12!} - \frac{x^{14}}{14!}. $$