Radius of convergance complex power series

In summary, the radius of convergence for a complex power series is the distance from the center of the series to the point at which the series converges. It is calculated using the ratio test, and it is significant because it determines the domain of convergence for the series. The radius of convergence cannot be negative, and changing the center of the series does not affect its value.
  • #1
Nikitin
735
27

Homework Statement



https://scontent-a.xx.fbcdn.net/hphotos-ash3/1390611_10201748262844961_2141774184_n.jpg

I need help with 7b. Theorem 3 = termwise differentiation and theorem 4 = termwise integration.

Homework Equations


The Attempt at a Solution



I have no idea how differentiation or integration would help me. Can somebody post a short strategy I can use? Please?
 
Last edited:
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  • #2
Termwise integration gives a term to "cancel" the n in the numerator (not exactly, however). That's not really necessary, but at least it uses integration.
 
  • #3
Hey! I solved it after looking at the problem with a fresh mind. Thx for the comment, tho
 

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

The radius of convergence for a complex power series is the distance from the center of the series to the point at which the series converges. It is represented by the letter R and can be found by using the ratio test on the series.

How is the radius of convergence calculated?

The radius of convergence can be calculated using the ratio test, which involves taking the limit of the absolute value of the ratio of the (n+1)th term to the nth term of the series. If this limit is less than 1, the series converges, and the radius of convergence is equal to the distance from the center to the point at which the series converges. If the limit is greater than 1, the series diverges, and the radius of convergence is 0. If the limit is exactly 1, further testing is needed to determine the convergence or divergence of the series.

What is the significance of the radius of convergence?

The radius of convergence is significant because it tells us the domain of convergence for the complex power series. Any value within the radius of convergence will result in a convergent series, while any value outside of the radius will result in a divergent series. This information is important in determining the validity and usefulness of the series in mathematical calculations.

Can the radius of convergence be negative?

No, the radius of convergence cannot be negative. It is always a positive value or 0. This is because the distance from the center of the series to the point at which the series converges cannot be negative.

How does changing the center of a complex power series affect the radius of convergence?

Changing the center of a complex power series does not affect the radius of convergence. The radius of convergence depends on the coefficients and powers of the terms in the series, and these do not change when the center is shifted. However, the interval of convergence may change, as the series may converge or diverge at different points for different centers.

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