If we use one of the established MacLaurin series to determine the power series for a similar function (eg. e^2x^2), will the interval of convergance always be the same as the original maclaurin ( eg. e^x)? or is it neccessary to compute the the interval again for our new series?
What about in the case when we multiply a simple power series by a polynomial or real number? Does the interval always stay the same as the original?
I hope that made sense.
First question: no, in general, the interval of convergence need no be the same as the original Maclaurin. Denoting f(x)=ex, we have that e2x2=f(2x2), so that the role of x is now taken by 2x2. This is a hint. At the same time I answered your second question: you do need to compute the interval of convergence of the new series, but that does not have to be done from scratch.
Multiplication by powers of x or by real numbers does not affect the interval of convergence.
Alright, I see. Thank you.