Solved examples: Term-by-term differentiation and integration of power series.

📚 Quick Study Guide

• 🔍 Power Series Representation: A power series centered at $a$ has the form $\sum_{n=0}^{\infty} c_n(x-a)^n$, where $c_n$ are coefficients and $x$ is a variable.
• 📏 Radius of Convergence: Every power series has a radius of convergence $R$. Inside the interval $(a-R, a+R)$, the series converges. Outside, it diverges. Convergence at the endpoints $a-R$ and $a+R$ must be checked separately.
• 📈 Term-by-Term Differentiation: If $f(x) = \sum_{n=0}^{\infty} c_n(x-a)^n$ converges for $|x-a| < R$, then $f'(x) = \sum_{n=1}^{\infty} nc_n(x-a)^{n-1}$ also converges for $|x-a| < R$.
• 📉 Term-by-Term Integration: If $f(x) = \sum_{n=0}^{\infty} c_n(x-a)^n$ converges for $|x-a| < R$, then $\int f(x) dx = C + \sum_{n=0}^{\infty} c_n \frac{(x-a)^{n+1}}{n+1}$ also converges for $|x-a| < R$, where $C$ is the constant of integration.
• 💡 Important Note: Term-by-term differentiation and integration do not change the radius of convergence. However, they *can* change the convergence behavior at the endpoints of the interval of convergence.

Practice Quiz

1. What is the derivative of the power series $\sum_{n=0}^{\infty} x^n$?

   1. $\sum_{n=0}^{\infty} nx^{n-1}$
   2. $\sum_{n=1}^{\infty} nx^{n-1}$
   3. $\sum_{n=0}^{\infty} x^{n+1}$
   4. $\sum_{n=1}^{\infty} x^{n+1}$

2. What is the integral of the power series $\sum_{n=0}^{\infty} x^n$?

   1. $C + \sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1}$
   2. $\sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1}$
   3. $C + \sum_{n=1}^{\infty} \frac{x^{n+1}}{n+1}$
   4. $\sum_{n=1}^{\infty} \frac{x^{n+1}}{n}$

3. If a power series has a radius of convergence $R=3$, what is the radius of convergence of its derivative?

   1. $R = 3$
   2. $R = 9$
   3. $R = 1$
   4. Cannot be determined.

4. If $f(x) = \sum_{n=0}^{\infty} c_n x^n$, what is $f'(0)$?

   1. $c_0$
   2. $c_1$
   3. $0$
   4. Cannot be determined.

5. What happens to the interval of convergence when integrating a power series?

   1. It always stays the same.
   2. It always gets smaller.
   3. It can change at the endpoints.
   4. It always gets larger.

6. The power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ represents which function?

   1. $\sin(x)$
   2. $\cos(x)$
   3. $e^x$
   4. $\ln(x)$

7. What is the derivative of $\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$?

   1. $\sum_{n=0}^{\infty} \frac{x^{2n-1}}{(2n-1)!}$
   2. $\sum_{n=1}^{\infty} \frac{x^{2n-1}}{(2n-1)!}$
   3. $\sum_{n=0}^{\infty} \frac{2nx^{2n-1}}{(2n)!}$
   4. $\sum_{n=1}^{\infty} \frac{2nx^{2n-1}}{(2n)!}$