# 🎯 Introduction to Power Series ## What is a Power Series? A **power series** is an infinite series of the form: $$\sum_{n=0}^{\infty} c_n (x - a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3 + \cdots$$ where: - $c_n$ are **coefficients** (constants) - $x$ is a **variable** - $a$ is the **center** > **💡 Key Idea**: A power series is like a polynomial with infinitely many terms! --- ## Power Series Centered at $a = 0$ When $a = 0$, we get a **Maclaurin-type power series**: $$\sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots$$ This is the most common form! --- ## Example 1: Geometric Series as Power Series The geometric series: $$\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots$$ This is a power series with $c_n = 1$ for all $n$, centered at $a = 0$. **Converges when**: $|x| < 1$ **Sum**: $\frac{1}{1-x}$ (when $|x| < 1$) --- ## Power Series as Functions A power series defines a **function**: $$f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n$$ The domain of $f$ is the set of all $x$ values where the series converges. --- ## Example 2: Recognize the Function What function does $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ represent? This is the **Taylor series for $e^x$**! $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$ **Converges for all $x$** (we'll prove this later). --- ## Convergence of Power Series For a given power series $\sum c_n(x-a)^n$, exactly **one** of the following is true: 1. **Converges only at $x = a$** (trivial case) 2. **Converges for all $x$** (radius of convergence $R = \infty$) 3. **Converges for $|x - a| < R$** and diverges for $|x - a| > R$ where $R$ is the **radius of convergence**. --- ## The Radius of Convergence The **radius of convergence** $R$ is: - $R = 0$ if series only converges at center - $0 < R < \infty$ if series converges on an interval - $R = \infty$ if series converges for all $x$ **Interval of convergence**: $(a - R, a + R)$ or $[a - R, a + R]$ or variations (Must check endpoints separately!) --- ## Finding Radius of Convergence **Use the Ratio Test**: $$\frac{1}{R} = \lim_{n \to \infty} \left|\frac{c_{n+1}}{c_n}\right|$$ Or equivalently: $$R = \lim_{n \to \infty} \left|\frac{c_n}{c_{n+1}}\right|$$ (If the limit exists or is $\infty$) --- ## Example 3: Find Radius Find the radius of convergence for $\sum_{n=0}^{\infty} \frac{x^n}{n+1}$. **Use Ratio Test**: Let $a_n = \frac{x^n}{n+1}$ $$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \left|\frac{\frac{x^{n+1}}{n+2}}{\frac{x^n}{n+1}}\right|$$ $$= \lim_{n \to \infty} \left|x \cdot \frac{n+1}{n+2}\right|$$ $$= |x| \cdot \lim_{n \to \infty} \frac{n+1}{n+2} = |x| \cdot 1 = |x|$$ --- For convergence: $|x| < 1$ **Radius of convergence**: $R = 1$ --- ## Example 4: Power Series Centered at $a = 2$ Find the radius of convergence for $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}$. **Use Ratio Test**: Let $a_n = \frac{(x-2)^n}{n \cdot 3^n}$ $$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \left|\frac{(x-2)^{n+1}}{(n+1) \cdot 3^{n+1}} \cdot \frac{n \cdot 3^n}{(x-2)^n}\right|$$ $$= \lim_{n \to \infty} \left|\frac{x-2}{3} \cdot \frac{n}{n+1}\right|$$ $$= \frac{|x-2|}{3} \cdot 1 = \frac{|x-2|}{3}$$ --- For convergence: $\frac{|x-2|}{3} < 1$ $$|x - 2| < 3$$ **Radius of convergence**: $R = 3$ **Center**: $a = 2$ **Interval** (before checking endpoints): $(2-3, 2+3) = (-1, 5)$ --- ## Operations on Power Series If $f(x) = \sum c_n x^n$ and $g(x) = \sum d_n x^n$ both converge for $|x| < R$: ### Addition/Subtraction: $$f(x) \pm g(x) = \sum (c_n \pm d_n) x^n$$ ### Scalar Multiplication: $$k \cdot f(x) = \sum (k \cdot c_n) x^n$$ ### Multiplication by $x^m$: $$x^m f(x) = \sum c_n x^{n+m}$$ --- ## Differentiation of Power Series **Theorem**: If $f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n$ with radius $R > 0$, then: $$f'(x) = \sum_{n=1}^{\infty} n \cdot c_n (x-a)^{n-1}$$ **Differentiate term by term!** The derivative has the **same radius of convergence** $R$. --- ## Example 5: Differentiate Power Series Find the derivative of $f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}$. **Differentiate term by term**: $$f'(x) = \sum_{n=1}^{\infty} \frac{n \cdot x^{n-1}}{n!} = \sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1)!}$$ Reindex with $m = n - 1$: $$= \sum_{m=0}^{\infty} \frac{x^m}{m!}$$ **Notice**: $f'(x) = f(x)$! This confirms $f(x) = e^x$ (since $\frac{d}{dx}[e^x] = e^x$). --- ## Integration of Power Series **Theorem**: If $f(x) = \sum_{n=0}^{\infty} c_n (x-a)^n$ with radius $R > 0$, then: $$\int f(x)\,dx = C + \sum_{n=0}^{\infty} \frac{c_n}{n+1}(x-a)^{n+1}$$ **Integrate term by term!** The integral has the **same radius of convergence** $R$. --- ## Example 6: Integrate Power Series Find $\int \sum_{n=0}^{\infty} x^n\,dx$ for $|x| < 1$. **Integrate term by term**: $$\int \sum_{n=0}^{\infty} x^n\,dx = C + \sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1}$$ We know $\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ for $|x| < 1$. So: $$\int \frac{1}{1-x}\,dx = -\ln|1-x| + C$$ This matches: $-\ln(1-x) = \sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1} = \sum_{n=1}^{\infty} \frac{x^n}{n}$ --- ## Uniqueness of Power Series **Theorem**: If $\sum c_n(x-a)^n = \sum d_n(x-a)^n$ for all $x$ in some interval, then: $$c_n = d_n \text{ for all } n$$ **Power series representation is unique!** --- ## Example 7: Find Coefficients If $\sum_{n=0}^{\infty} c_n x^n = \frac{1}{1-x}$ for $|x| < 1$, find $c_n$. We know: $\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots$ By uniqueness: $c_n = 1$ for all $n \geq 0$. --- ## Substitution in Power Series Can substitute $u = f(x)$ into a power series! **Example**: If $\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ for $|x| < 1$: Replace $x$ with $-x^2$: $$\sum_{n=0}^{\infty} (-x^2)^n = \frac{1}{1-(-x^2)} = \frac{1}{1+x^2}$$ $$1 - x^2 + x^4 - x^6 + \cdots = \frac{1}{1+x^2}$$ (for $|x| < 1$) --- ## ⚠️ Common Mistakes ### Mistake 1: Forgetting Absolute Value **WRONG**: $|x - 2| < 3$ means $x < 5$ **RIGHT**: $|x - 2| < 3$ means $-3 < x - 2 < 3$, so $-1 < x < 5$ --- ### Mistake 2: Not Checking Endpoints Radius tells you interval is $(a-R, a+R)$, but series might converge at one or both endpoints! Must check $x = a - R$ and $x = a + R$ separately. --- ### Mistake 3: Wrong Index After Differentiation When differentiating $\sum_{n=0}^{\infty} c_n x^n$: Get $\sum_{n=1}^{\infty} n c_n x^{n-1}$ (starts at $n=1$, not $n=0$!) The $n=0$ term disappears (constant → 0). --- ### Mistake 4: Confusing Center and Radius For $\sum c_n(x-2)^n$ with $R = 3$: - **Center**: $a = 2$ - **Radius**: $R = 3$ - **Interval**: $(-1, 5)$ before checking endpoints Not centered at 0! --- ## 📝 Practice Strategy 1. **Identify center $a$**: Look at $(x - a)^n$ 2. **Use Ratio Test**: Find $\lim|\frac{a_{n+1}}{a_n}|$, set $< 1$ for convergence 3. **Solve for $x$**: Get $|x - a| < R$ 4. **Check endpoints**: Test $x = a \pm R$ separately using other tests 5. **For differentiation**: Differentiate each term, start sum at $n=1$ 6. **For integration**: Integrate each term, add constant 7. **Known series**: Memorize $\frac{1}{1-x} = \sum x^n$ and $e^x = \sum \frac{x^n}{n!}$