Question

How do you find the power series representation of \[\dfrac{{\arctan \left( x \right)}}{x}\] and what is the radius of convergence?

Answer 1

Hint: In the given question, we have been given an expression whose power series representation is to be solved for. Then we have to find the radius of convergence. We are going to solve it by first finding the power series representation of the numerator and then dividing it by the one of the denominator. Then we are going to apply the formula of radius of convergence so as to complete the question’s answer.

Complete step by step solution:
In the given question, we have to find the power series representation of \[\dfrac{{\arctan \left( x \right)}}{x}\].
We know, \[\arctan {\left( x \right)^\prime } = \dfrac{1}{{1 + {x^2}}} = \sum\limits_n {{{\left( { - 1} \right)}^n}{x^{2n}}} \].
Now, using the standard result, we know, the power series representation of \[\dfrac{1}{{1 - x}}\] is \[\sum\limits_n {{x^n}} \] such that \[\left| x \right| < 1\].
Hence, the power series representation of \[\arctan \left( x \right)\] is
\[\int {\sum\limits_n {{{\left( { - 1} \right)}^n}{x^{2n}}dx} } = \sum\limits_n {\dfrac{{{{\left( { - 1} \right)}^n}}}{{2n + 1}}{x^{2n + 1}}} \]
Now, we divide this whole expression by \[x\] to find the power series representation of \[\dfrac{{\arctan \left( x \right)}}{x}\].
So, we have,
the power series representation of \[\dfrac{{\arctan \left( x \right)}}{x}\] is \[\sum\limits_n {\dfrac{{{{\left( { - 1} \right)}^n}}}{{2n + 1}}{x^{2n}}} \].
Now, let \[\sum\limits_n {\dfrac{{{{\left( { - 1} \right)}^n}}}{{2n + 1}}{x^{2n}} = u} \].
For finding the radius of convergence, we find,
\[\mathop {\lim }\limits_{n \to + \infty } \left| {\dfrac{{{u_{n + 1}}}}{{{u_n}}}} \right|\]
Now, \[\dfrac{{{u_{n + 1}}}}{{{u_n}}} = {\left( { - 1} \right)^{n + 1}}\dfrac{{{x^{2n + 2}}}}{{2n + 3}}\dfrac{{2n + 1}}{{{{\left( { - 1} \right)}^n}{x^{2n}}}} = - \dfrac{{2n + 1}}{{2n + 3}}{x^2}\]
Hence, \[\mathop {\lim }\limits_{n \to + \infty } \left| {\dfrac{{{u_{n + 1}}}}{{{u_n}}}} \right| = \left| {{x^2}} \right| = {x^2}\]

Note: In the given question, we were given an expression whose power series representation and radius of convergence was to be found. To solve this question, it is a necessity that we have a very deep and firm knowledge of the subject. Without that, all our work goes in vain. Here, we needed to know the exact formulae and representations of the given expression.