Series: f(x)=10x^2 arctan(x^6) is sum[n=0,infty](c_n)(x^n)

[matt3D]

Hello, I have a problem that says:

The function

is represented as a power series

What is the lowest term with a nonzero coefficient and find the radius of convergence R of the series.

Okay so, I know that:

\(\displaystyle 10x^2\sum_{n=0}^\infty(-1)^n \frac{(x^6)^{2n+1}}{2n+1}\)

The problem is I'm not sure why I would use that and how I would go through the rest of the problem, my teacher went over a similar problem and I didn't understand what she meant. I know that the radius of convergence is 1 since \(\displaystyle \| x^6 \| < 1\). I'm not sure why there are two absolute bars, I only wanted one set but I couldn't get latex to just put one on each side... :mrgreen:

Thanks,
Matt

 

[Deleted member 4993]

Hello, I have a problem that says:

The function

is represented as a power series

What is the lowest term with a nonzero coefficient and find the radius of convergence R of the series.

Okay so, I know that:

\(\displaystyle 10x^2\sum_{n=0}^\infty(-1)^n \frac{(x^6)^{2n+1}}{2n+1}\)

The problem is I'm not sure why I would use that and how I would go through the rest of the problem, my teacher went over a similar problem and I didn't understand what she meant. I know that the radius of convergence is 1 since \(\displaystyle \| x^6 \| < 1\). I'm not sure why there are two absolute bars, I only wanted one set but I couldn't get latex to just put one on each side... :mrgreen:

Thanks,
Matt

Have you worked with Taylor series?

You are looking at Taylor series expansion of "arctan" function.