taylor series of tan^-1(x)/x

[taipenx]

taylor series of (tan^-1(x))/x

i am lost on how to do this an explanation would be much appreciated

 

[mario99]

You want to find the Taylor series of $\displaystyle g(x) = \frac{\tan^{-1}x}{x}$.

First question comes to mind, can we reduce this problem to a smaller one? Yes. We want to find the Taylor series of $\displaystyle f(x) = \tan^{-1}x$, then when we are done we will divide the series by $\displaystyle x$.

If you want to find the Taylor series of a function, you have to tell about what point this series will be. Otherwise, these mathematicians will assume a point that you don't like. And because I am not a mathematician, I will assume a peaceful point, $\displaystyle x = 0$, which happens to be that we are finding the Maclaurin series of $\displaystyle f(x)$.

Don't worry about the name. The Maclaurin series is just a Taylor series about $\displaystyle x = 0$.

Now comes the difficult (fun) part. We will introduce the machine that produces the Maclaurin series:

$\displaystyle \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$

This notation is a little difficult to understand, so I will simplify it.

$\displaystyle \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdot\cdot\cdot \ \cdot$

Basically, you just need to take the derivatives of the function $\displaystyle f(x) = \tan^{-1}x$ and plug them above after substituting $x = 0$. But this machine produce infinite derivatives! Now is the important question, how many of them do you need? The answer is, until you recognize a pattern. Don't panic, usually $2$ to $3$ derivatives will be enough to see the pattern. Try it!

After you recognize the pattern, you will create an infinite series.

$\displaystyle \tan^{-1}x = \sum_{n=0}^{\infty}a_nx^{k(n)}$

The final step is to divide by $x$:

$\displaystyle \frac{\tan^{-1}x}{x} = \sum_{n=0}^{\infty}\frac{a_nx^{k(n)}}{x} = \sum_{n=0}^{\infty}a_nx^{k(n)-1}$

At this point, you have successfully found the Taylor series of $g(x)$ about $x = 0$.


Bonus question. Does this series converge for all values of $x$?

 

[khansaheb]

taylor series of (tan^-1(x))/x

i am lost on how to do this an explanation would be much appreciated

Are you following a class/textbook?