- #1
Niles
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Homework Statement
Hi all.
Lets say I have a complex power series given by
[tex]
\sum_{n=0}^\infty c_nz^n,
[/tex]
where z is a complex number and c is a complex constant. Inside its circle of convergence, I can differentiate it leading to
[tex]
\sum_{n=0}^\infty c_nnz^{n-1} = \sum_{n=0}^\infty (n+1)c_{n+1}nz^{n}.
[/tex]
If I want to find the circle of convergence for this series, then I can use the ratio test
[tex]
\frac{1}{R} = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 1} }}{{a_n }}} \right|.
[/tex]
I have two questions for this:
Question #1: Does the ratio test give me the same answer regardless of I substitute e.g. n -> n+3? I.e., is it correct that
[tex]
\mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 1} }}{{a_n }}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{a_{n + 3} }}{{a_{n + 2} }}} \right|.
[/tex]
Personally, I think it does not matter, because we let n go to infinity.
Question #2: Does the ratio test only work for power series that go from n=0 to infinity, or do they also work if n start at e.g. 1 or -1?Thank you very much in advance.Niles.