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What is the Maclaurin series for the function f(x)=arcsinx. Find the radius of convergence of the...

What is the Maclaurin series for the function f(x)=arcsinx. Find the radius of convergence of the series.

Expert Solution

Solution:

Given,

Taking the first derivative:

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 can integrate term by term, to gain the power series for arcsin(x):

It can also be write as

Applying ratio test:

Therefore Radius of convergence

Convergence interval is,

milcah answered 3 years ago

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