- #1

- 1,868

- 0

## 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.

Best regards,

Niles.