I am required to find the radius of convergence for the
function $$f(x) = 5x^3 – 6x^2 – 7x + 6$$ by first finding its Maclaurin series.
I found the Maclaurin series to be $$ 6 – 7x -6x^2 + 5x^3…$$
I am having trouble figuring out what the summation form notation of this series should look like. I would need this to apply the ratio test and find the radius of convergence.
Some guidelines on figuring out the summation notation for any such series would also be appreciated. Thanks!
Best Answer
The coefficients on the $x^4$ term and beyond are all zero so the ratio test isn't suitable here. On the other hand you can use the root test to show the radius of convergence is infinite. That is, the finite series converges to $f$ everywhere on the line.
Looking back, the root test isn't really necessary. A finite power series is clearly convergent at every point. To what? Itself.