# Math Help - radius of convergence

1. ## radius of convergence

how come when u multiply a series by x^n or when u multiply a series by a constant, it doesnt change the radius of convergence? ex: the power series representation for e^x, the ROC doesnt change for the series represenation of x^2e^x, or if a constant was multiplied to the series represenation for e^x, i know that if u use the ratio test it's true that the ROC doesnt change, but like can someone explain to me like why intuitively it would be true?

2. What we're doing doing is a limit comparison to the geometric series.
That's why we compare the next term to our present term.

And since $\biggl({x+1\over x}\biggr)^2\to 1$ as $x\to\infty$

theterms of the form $x^2$ can't change the interior of the interval of convergence.
However, it can possibly alter convergence/divergence at the endpoints.