Rules regarding power series and radii of convergence...?

Question

My first question is if I were to multiply a power series expansion of a known function by another known function (of which I don't know the power series expansion), is it acceptable to multiply every term in the power series by this function (because it's like a polynomial)?

Perhaps stated a bit better, if I have a function: f(x) = sum[(a_n)(x^n)] and I want to multiply it by say, x^2, can I do this: (x^2)f(x) = (x^2)sum[(a_n)(x^n)]
(x^2)f(x) = sum[(x^2)(a_n)(x^n)]
(x^2)f(x) = sum[(a_n)(x^[n+2])]

And my second question is if I have a power series with a negative exponent, say sum[x^(-2)], can I even consider it a power series? I ask this because I've looked and looked and only found examples of power series that have x^n or x^2n. And if that is a power series, how could I begin to find the radius of convergence?

Any help will be appreciated.

Note: sum[] of course means summation notation from n = 0 to infinity.

In the second question, where I have sum[x^(-2)], I meant sum[x^(-2n)].

Answer 1

For the first one: yes, you can multiply the expression term by term. If f(x) is a function given by [n=0, ∞]∑(a_n x^n), and g(x) is any function, then f(x) * g(x) = [n=0, ∞]∑(g(x) a_n x^n). g(x) is just a constant in terms of n, so distributing it across the terms of the series will not change its convergence properties.

For the second question: you seem to be describing a Laurent series. These series are described in more detail on the Wikipedia:

http://en.wikipedia.org/wiki/Laurent_series

Answer 2

Eighteight^(n+a million) * 8^n / eightx?8eight8^n / 8^(n+a million) . . . . . . . . . . . . . = eighteight series converges for all fee of x such that 88x?eight8 -----> radius of convergence = 8 ?8 < x?8 < 8 0 < x < sixteen Now we confirm endpoints: x = 0 ?[n=a million to ?] (x?8)^n / 8^n = ? (?8)^n / 8^n = ? (?a million)^n = ?a million + a million ? a million + a million ? . . . This diverges x = sixteen ?[n=a million to ?] (x?8)^n / 8^n = ? 8^n / 8^n = ? a million = a million + a million + a million + a million + . . . This diverges era convergence: (0, sixteen) Radius of convergence: 8