Finding the Radius of Convergence for a Complex Function.

In summary: SorryIn summary, the radius of convergence for the Taylor expansion of f(z) is determined by the distance between the point of expansion and the closest singularity, which is -1, 1, 2, and 3 in this case. By expanding each factor separately and multiplying the series together, a lower bound for the radius of convergence can be found.
  • #1
shedrick94
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< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

How would you find the radius of convergence for the taylor expansion of:

\begin{equation} f(z)=\frac{e^z}{(z-1)(z+1)(z-3)(z-2)} \end{equation}

I was thinking that you would just differentiate to find the taylor expansion and then use the ratio test but this seems far too tedious to be the right way to do it! Any help?
 
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  • #2
The Taylor expansion of ez converges for all z. The denominator introduces poles at -1, 1, 2 and 3, so you need to be sufficiently far away from those values. Now determine "sufficiently far away"...
 
  • #3
shedrick94 said:
< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

How would you find the radius of convergence for the taylor expansion of:

\begin{equation} f(z)=\frac{e^z}{(z-1)(z+1)(z-3)(z-2)} \end{equation}

I was thinking that you would just differentiate to find the taylor expansion and then use the ratio test but this seems far too tedious to be the right way to do it! Any help?

Should we assume you want to expand around ##z = 0##?
 
  • #4
shedrick94 said:
< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

How would you find the radius of convergence for the taylor expansion of:

\begin{equation} f(z)=\frac{e^z}{(z-1)(z+1)(z-3)(z-2)} \end{equation}

I was thinking that you would just differentiate to find the taylor expansion and then use the ratio test but this seems far too tedious to be the right way to do it! Any help?

If (as I asked in #3 but did not receive an answer!) you are expanding about ##z = 0##, you can just expand each factor separately, then multiply the series together. The resulting series will be messy and not easy to write explicitly, but at least you can say something about the radius of convergence, since a lower bound on the radius of convergence of a product of series is known in terms of the individual radii of convergence. Google 'product of power series' or something similar.
 
  • #5
Sorry I understand this now. The expansion was about z=i, but I understand you would just the distance between the place you are expanding around and the closest singularity.
 

What is the Radius of Convergence?

The Radius of Convergence is a mathematical concept used to determine the values of a variable for which an infinite series will converge. It is represented as a positive number, denoted by R, and is used to determine the range of values for which a series will converge.

How is the Radius of Convergence calculated?

The Radius of Convergence is calculated using the ratio test, which compares the absolute value of each term in the series to the absolute value of the previous term. The ratio of these values is then taken as the variable approaches infinity, and the limit is calculated. The Radius of Convergence is equal to the reciprocal of this limit.

Why is the Radius of Convergence important?

The Radius of Convergence is important because it helps determine the convergence or divergence of an infinite series. It also allows for the determination of the interval of convergence, which is the range of values for which the series will converge.

What happens if the Radius of Convergence is zero?

If the Radius of Convergence is zero, the infinite series will not converge for any value of the variable. This means that the series diverges and does not have a finite sum.

Can the Radius of Convergence be negative?

No, the Radius of Convergence cannot be negative as it represents the distance from the center of convergence on a complex plane. It can only be a positive value or infinity.

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