1. If A ⊆ , define −A to be {−a: a ∈ A}. Let A be the interval of convergence of the series Σa n x n . Prove that the interval of convergence of Σ(−1) n a n x n is − A. 2. Suppose that the series Σa n...

1. If A ⊆
, define −A to be {−a: a ∈ A}. Let A be the interval of convergence of the series Σa_nxⁿ. Prove that the interval of convergence of Σ(−1)ⁿa_nxⁿ
is − A.

2. Suppose that the series Σa_nxⁿ
has radius of convergence 2. Find the radius of convergence of each series, where k is a fixed positive integer

3. Prove that the series   Σ^∞
_n=oa_nxⁿ
and  Σ^∞
_n=0
na_nxⁿ
have the same radius of convergence (finite or infinite