# What are some 'general solutions' for infinite series?

For example we have the general solution for the infinite series for computing natural logs:

$$\ln(x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}(x-1)^n$$

Where we can input any whole integer value for x and the series will give us exactly what the ln of that number is equal to, but aside from this I have been unable to find anything else. I am not strictly speaking of natural logs but general solutions of any type.

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I'm pretty sure that series does not give you the value of $\ln (x)$ for any whole integer value of $x$. – David Mitra Feb 14 '14 at 18:55
I really have no idea what you're asking. Do you want more examples of functions which can be calculated using infinite series? – Jack M Feb 14 '14 at 18:57
your formula is incorrect, but also your question is not clear enough – Sameh Shenawy Feb 14 '14 at 19:08

If you're asking for examples of Taylor Series:

\begin{align*} e^x &= \sum_{n=0}^\infty \frac{x^n}{n!} \\ \sin x &= \sum_{n=0}^\infty \left(-1\right)^n\frac{x^{2n+1}}{\left(2n+1\right)!}& x\in\left[-\pi,\pi\right]\\ \cos x &= \sum_{n=0}^\infty \left(-1\right)^n\frac{x^{2n}}{\left(2n\right)!}& x\in\left[-\pi,\pi\right] \end{align*}

You can construct a series representation for any function by taking its Taylor series. (However, some functions may not yield interesting expansions: for example, the Taylor series of any polynomial function will terminate and reduce to the function itself.)

A handy tool is WolframAlpha. For example, the query "series expansion ln(x)" yields this page, which gives a number of expansions, including the one in your question.

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Hah, I totally forgot about those. Sometimes there is such a thing as being too focused on a task :) – eatscrayons Feb 14 '14 at 20:18

Taylor Series of any function will do, like the one you displayed.

That Wiki article lists a ton of examples.

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Sure does, thanks for the link! – eatscrayons Feb 14 '14 at 20:28