### Radius of Convergence for Power Series**4.** If $\sum a_n z^n$ has a radius of convergence $R$, what is the radius of convergence of each of the following?**(a)** $\sum a_n z^{2n}$**(b)** $\sum a_n z^{n/2}$**5.** Find the radius of convergence for each of the following series.---**Explanation:**The problem involves determining the radius of convergence for variations of a given power series. Given that a power series $\sum a_n z^n$ has a certain radius of convergence $R$, you need to find out how the radius of convergence changes when the arguments of the series are altered.For part **(a)**, the argument is adjusted to $z^{2n}$, and for part **(b)**, the argument is changed to $z^{n/2}$. Understanding how transformations of the series' arguments affect the radius of convergence requires an understanding of series manipulations and transformations in complex analysis.

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4. If a,z" following? has a radius of convergence R. what is the radius of convergence of each of the (a) an2 2.2m

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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### Radius of Convergence for Power Series

**4.** If $\sum a_n z^n$ has a radius of convergence $R$, what is the radius of convergence of each of the following?

**(a)** $\sum a_n z^{2n}$

**(b)** $\sum a_n z^{n/2}$

**5.** Find the radius of convergence for each of the following series.

---

**Explanation:**

The problem involves determining the radius of convergence for variations of a given power series. Given that a power series $\sum a_n z^n$ has a certain radius of convergence $R$, you need to find out how the radius of convergence changes when the arguments of the series are altered.

For part **(a)**, the argument is adjusted to $z^{2n}$, and for part **(b)**, the argument is changed to $z^{n/2}$. Understanding how transformations of the series' arguments affect the radius of convergence requires an understanding of series manipulations and transformations in complex analysis.

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