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Maclaurin Series

The Maclaurin series for the inverse hyperbolic tangent is of the form x+x^3/3+x^5/5...x^7/7. Show that this is true through the third derivative term.

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Posting # MATH 8689

The problem asks us to find the Maclaurin series expansion of the function
f(x) = tanh-1x

Now, in general the Maclaurin series for any function is:

where f'(x) , f"(x), f(3)(x) ... f (n)(x) represent the first, second , third and nth derivatives of f(x) resp. and the Maclaurin series is expanded around the origin i.e. the derivatives are evaluated at x=0. We will simply use the above formula to deduce the answer.

Now let us look at the function f (x) = tanh-1x more closely and evaluate each of the above terms. The problem just asks us to go till the 3rd derivative so we shall stop at f(3)(x).

(i) 1st term of the formula is: f(0) . This means simply that ...

Solution Summary

A relation involving a Maclaurin series is proven. Hyperbolic tangents are analyzed. The solution is detailed and well presented.