Question

Compute the Taylor series at x = 0 for ln(1+x) and for x cos x by repeatedly differentiating the function. Find the radii of convergence of the associated series.

Answer 1

Taylor series at x = 0 is called Maclaurin series.

Formula is,

For

  

So on.

So we can write

For

  

So on

So we can write

Now we need to find the radii of convergence of these series.

For that we can use ratio test.

then radius of convergence is

For first function

So on applying limits we get

To find convergence of we need to ratio test,

Series is convergent with radius of convergence equal to infinity

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