# How do you find the Maclaurin Series for #arcsin(x)/x#?

# arcsinx/x = 1 + 1/6x^2 + 3/40 x^4 + 5/112 x^6 + 35/1162 x^8 + ... #

We seek a Maclaurin series for the function:

We would generally form a general Maclaurin series using:

So we would take:

and form the first derivative (from tables):

Along with higher derivatives. But this method quickly gets cumbersome, and in fact there is a much faster way of generating the series we need, by using the result [A] whose Maclaurin series we can actually generate very quickly using the the binomial series:

As follows:

Then, we have:

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