Finding Radius of Convergence of the Power Series

In summary, the conversation discussed finding the radius of convergence of a power series for a function expanded about a given point. The relevant equation was determined to be the ratio test, and it was mentioned that the power series should be defined for all z near the given point. The only value of z that may cause problems is the point that is sqrt(2) away from the given point.
  • #1
brianhawaiian
12
0

Homework Statement


"Find the radius of convergence of the power series for the following functions, expanded about the indicated point.

1 / (z - 1), about z = i.



Homework Equations



1 / (1 - z) = 1 + z + z^2 + z^3 + z^4 + ... +

Ratio Test: limsup sqrt(an)^k)^1/k



The Attempt at a Solution



1 / (z - 1) = -1 / (1 - z) = -1(1 + z + z^2 + z^3 + z^4 + ... +)

Ratio test gives 1?

Answer but I'm not sure how sqrt(2)
 
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  • #2
Your relevant equation is irrelevant. Your series is not about z = 0 - it's about z = i, so your power series will be in powers of z - i, not powers of z. So your power series should be defined for all z near i. The only value of z where you'll run into problems is z = 1, which happens to be sqrt(2) away from i. Hope that helps.
 

Related to Finding Radius of Convergence of the Power Series

1. What is the radius of convergence of a power series?

The radius of convergence of a power series is the distance from the center of the series to the nearest point where the series converges. It is a measure of how far the series can be extended before it diverges.

2. How do you find the radius of convergence of a power series?

To find the radius of convergence, we use the ratio test, which compares the absolute value of the terms in the series to a geometric series. If the limit of this ratio is less than 1, then the series converges and the radius of convergence is the value of x for which this limit is equal to 1. If the limit is greater than 1, then the series diverges.

3. What is the interval of convergence for a power series?

The interval of convergence is the range of values for x in which the power series converges. It is determined by the radius of convergence, with the interval being centered at the center of the series and extending to the left and right by the radius value.

4. Can the radius of convergence of a power series be negative?

No, the radius of convergence is always a positive value. It represents a distance and therefore cannot be negative.

5. How do you deal with endpoints when finding the interval of convergence?

At the endpoints of the interval, the series may converge or diverge. To determine this, we can use the endpoints test, which involves plugging in the value of x at the endpoint into the original series. If the series converges at the endpoint, then it is included in the interval of convergence. If the series diverges at the endpoint, then it is not included in the interval of convergence.

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