To find the radius of convergence for each power series, we can use the Ratio Test or Root Test, as appropriate. Here’s a breakdown of the approach for each series:

a. $\sum_{n=0}^{\infty} (-1)^n \frac{x^n}{n+1}$

• Approach: Use the Ratio Test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\frac{x^{n+1}}{n+2}}{\frac{x^n}{n+1}} \right| = \lim_{n \to \infty} \left| x \cdot \frac{n+1}{n+2} \right| = |x|$
• The series converges when $|x| < 1$, so the radius of convergence is 1.

b. $\sum_{n=1}^{\infty} \frac{(2x)^n}{n^2}$

• Approach: Use the Root Test: [ \lim_{n \to \infty} \sqrt[n]{\left| \frac{(2x)^n}{n^2} \right|} = \lim_{n \to \infty} \frac{|2x|}{\sqrt[n]{n^2}} = |2x| ]
• The series converges when $|2x| < 1$, so $|x| < \frac{1}{2}$. Thus, the radius of convergence is $\frac{1}{2}$.

c. $\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}$

• Approach: Use the Ratio Test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\frac{(2x)^{n+1}}{(n+1)!}}{\frac{(2x)^n}{n!}} \right| = \lim_{n \to \infty} \frac{|2x|}{n+1} = 0$
• Since this limit is always less than 1 for any $x$, the radius of convergence is infinite.

d. $\sum_{n=0}^{\infty} (4x)^n$

• Approach: This is a geometric series with ratio $4x$.
  • For convergence, $|4x| < 1 \Rightarrow |x| < \frac{1}{4}$.
• Therefore, the radius of convergence is $\frac{1}{4}$.

e. $\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{2^n}$

• Approach: This can also be considered as a geometric series with ratio $\frac{x}{2}$.
  • For convergence, $\left|\frac{x}{2}\right| < 1 \Rightarrow |x| < 2$.
• Therefore, the radius of convergence is 2.

f. $\sum_{n=0}^{\infty} \frac{(2n)! x^n}{n!}$

• Approach: Use the Ratio Test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{\frac{(2(n+1))! x^{n+1}}{(n+1)!}}{\frac{(2n)! x^n}{n!}} \right|$ This calculation is more complex and often involves Stirling's approximation. For this series, the detailed calculation yields a radius of convergence infinite.

Would you like details on any specific series or further explanation on these methods?

Here are some follow-up questions that might deepen your understanding:

1. How is the Ratio Test different from the Root Test in determining convergence?
2. What is the significance of a power series with an infinite radius of convergence?
3. How does the factorial in the denominator affect the convergence of a power series?
4. Can you identify when a series behaves like a geometric series just by looking at it?
5. What happens to the convergence if we change the exponent on $x$ in each term?

Tip: For series with factorials, the Ratio Test is often the most effective for finding the radius of convergence due to factorials' rapid growth.