Radius of convergence of complex power series

In summary, the radius of convergence for the series ∑ n^-1.z^n is |z|. This is determined by the ratio test for convergence, which shows that the series converges for |z| < 1 and diverges for |z| > 1. Additionally, the lemma provided can be used to show that all values of |z| less than the radius of convergence will also converge, while values greater than the radius of convergence will diverge.
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Homework Statement


Find the radius of convergence of the series:

∑ n^-1.z^n
n=1

Use the following lemma:
∞ ∞
If |z_1 - w| < |z_2 - w| and if ∑a_n.(z_2 - w)^n converges, then ∑a_n.(z_1 - w)^n also
n=1 n=1
converges.
The contrapositive is also true.

Homework Equations





The Attempt at a Solution


Hi, here's what I've done:

Let r be the radius of convergence. The series converges when |z-w| < r and diverges for |z-w|> r.

In this question, z = z, w = 0.

Use the ratio test for convergence:

lim |a_n+1 / a_n| = lim |zn / n+1 |
n->∞
= |z|

Thus the series converges when |z| < 1
diverges when |z| > 1

So let |z - 0| = |z_2 - w|
Then, by the lemma, all |z_1 - 0| converges, where |z_1 - 0| < |z - 0|
and the converse is true for a diverging series.

Thus the radius of convergence = |z|

---

I think I've covered everything, but I don't think I've made very satisfactory use of the lemma. Can anyone please point me in the right direction?
Thanks for any help.
 
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  • #2
Your final answer appears to be "Thus the radius of convergence is |z|" which is meaningless. the radius of convergence is a real number, not a variable.

You had already said "Thus the series converges when |z| < 1
diverges when |z| > 1". What more do you need?
 

What is the definition of radius of convergence for a complex power series?

The radius of convergence of a complex power series is a measure of the distance from the center of the series to the nearest point where the series is no longer convergent. It is denoted by the letter R and is equal to the reciprocal of the limit of the absolute value of the ratio of consecutive coefficients in the series.

How is the radius of convergence related to the convergence of a complex power series?

The radius of convergence determines the interval of values for which the series converges. If the value of x is within the radius of convergence, the series will converge and the sum of the series will approach a finite value. However, if the value of x is outside the radius of convergence, the series will diverge and the sum will approach infinity.

What is the significance of the radius of convergence in complex analysis?

The radius of convergence is an important concept in complex analysis as it allows us to determine the convergence of a complex power series and the region of convergence. It also helps us to understand the behavior of functions represented by power series and to approximate these functions within a certain interval.

How can the radius of convergence be calculated for a complex power series?

The radius of convergence can be calculated using various methods, including the ratio test, the root test, and the Cauchy-Hadamard theorem. These methods involve evaluating the limit of the ratio of consecutive coefficients or the limit of the nth root of the absolute value of the coefficients.

What are some applications of the radius of convergence in mathematics and science?

The radius of convergence has various applications in mathematics and science, particularly in complex analysis, differential equations, and numerical analysis. It is also used in physics and engineering to approximate solutions to problems and to model physical phenomena using power series. Additionally, the radius of convergence plays a crucial role in the development of algorithms for solving problems in these fields.

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