1) Introduction: The Power of Infinite Sums 📚
Infinite series are the sum of an infinite sequence of terms, and determining whether they converge to a finite value or diverge is a cornerstone of advanced calculus. This topic introduces infinite series and provides tools like convergence tests to analyze them. These concepts are fundamental in mathematics, physics (e.g., Fourier series), and engineering, enabling us to handle sums that go on forever.
We’ll cover:
- Definition of Series: Understanding series and partial sums.
- Convergence and Divergence: Identifying finite or infinite sums.
- Convergence Tests: Techniques like ratio, root, and integral tests.
- Special Series: Geometric and p-series examples.
Quick Recap: A sequence’s limit prepares us; now we sum its terms to form a series.
2) Definition of Infinite Series 🎓
An infinite series is the sum \( \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \cdots \). The partial sum \( S_n = \sum_{k=1}^n a_k \) approximates the series, and the limit of \( S_n \) as \( n \to \infty \) determines convergence.
Definition 24.1: Infinite Series
A series \( \sum a_n \) converges to \( S \) if \( \lim_{n \to \infty} S_n = S \), where \( S_n = a_1 + \cdots + a_n \). If the limit does not exist or is infinite, the series diverges.
Example: \( \sum_{n=1}^{\infty} \frac{1}{2^n} \).
Example 1: Partial Sums
Compute the first three partial sums of \( \sum_{n=1}^{\infty} \frac{1}{n(n+1)} \).
- \( S_1 = \frac{1}{1 \cdot 2} = \frac{1}{2} \)
- \( S_2 = \frac{1}{2} + \frac{1}{2 \cdot 3} = \frac{1}{2} + \frac{1}{6} = \frac{2}{3} \)
- \( S_3 = \frac{2}{3} + \frac{1}{3 \cdot 4} = \frac{2}{3} + \frac{1}{12} = \frac{3}{4} \)
Answer: \( S_1 = \frac{1}{2} \), \( S_2 = \frac{2}{3} \), \( S_3 = \frac{3}{4} \).
Example 2: Telescoping Series
Evaluate \( \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right) \).
- \( S_n = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right) = 1 - \frac{1}{n+1} \).
- Limit: \( \lim_{n \to \infty} \left(1 - \frac{1}{n+1}\right) = 1 \).
Answer: \( 1 \).
3) Convergence and Divergence 📐
A series converges if its partial sums approach a finite limit; otherwise, it diverges. The divergence test helps identify divergence.
Definition 24.2: Convergence Test
If \( \lim_{n \to \infty} a_n \neq 0 \) (or does not exist), \( \sum a_n \) diverges. If the limit is 0, the test is inconclusive.
Example 3: Divergence Test
Test \( \sum_{n=1}^{\infty} \frac{n}{n+1} \) for convergence.
- \( \lim_{n \to \infty} \frac{n}{n+1} = \lim \frac{1}{1 + \frac{1}{n}} = 1 \neq 0 \).
- By divergence test, the series diverges.
Answer: Diverges.
Example 4: Convergence Check
Test \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) using partial sums.
- \( \lim_{n \to \infty} \frac{1}{n^2} = 0 \), inconclusive.
- Partial sums suggest convergence (p-series, later).
Answer: Inconclusive, but converges (to \( \frac{\pi^2}{6} \), see p-series).
4) Convergence Tests 🔍
Several tests determine convergence, building on the divergence test.
Definition 24.3: Convergence Tests
- Ratio Test: \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \). Converges if \( L < 1 \), diverges if \( L > 1 \), inconclusive if \( L = 1 \). - Root Test: \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = L \). Converges if \( L < 1 \), diverges if \( L > 1 \), inconclusive if \( L = 1 \). - Integral Test: If \( a_n = f(n) \) where \( f \) is positive, continuous, decreasing, \( \sum a_n \) converges if \( \int_1^{\infty} f(x) \, dx \) converges.
Example 5: Ratio Test
Test \( \sum_{n=1}^{\infty} \frac{2^n}{n!} \) for convergence.
- \( a_{n+1} = \frac{2^{n+1}}{(n+1)!} \), \( a_n = \frac{2^n}{n!} \).
- \( \left| \frac{a_{n+1}}{a_n} \right| = \frac{\frac{2^{n+1}}{(n+1)!}}{\frac{2^n}{n!}} = \frac{2}{n+1} \).
- \( \lim_{n \to \infty} \frac{2}{n+1} = 0 < 1 \).
- Converges.
Answer: Converges.
Example 6: Integral Test
Test \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) for convergence.
- \( f(x) = \frac{1}{x^2} \), positive, continuous, decreasing for \( x \geq 1 \).
- \( \int_1^{\infty} \frac{1}{x^2} \, dx = \lim_{b \to \infty} \left[ -\frac{1}{x} \right]_1^b = 0 - (-1) = 1 \).
- Converges (finite).
Answer: Converges.
5) Special Series and Their Behavior 🔍
Certain series have predictable convergence, like geometric and p-series.
Definition 24.4: Special Series
- Geometric Series: \( \sum_{n=0}^{\infty} ar^n \) converges to \( \frac{a}{1-r} \) if \( |r| < 1 \), diverges if \( |r| \geq 1 \). - p-Series: \( \sum_{n=1}^{\infty} \frac{1}{n^p} \) converges if \( p > 1 \), diverges if \( p \leq 1 \).
Example 7: Geometric Series
Determine if \( \sum_{n=0}^{\infty} \frac{3}{4^n} \) converges and find its sum.
- \( a = 3 \), \( r = \frac{1}{4} < 1 \).
- Sum: \( \frac{3}{1 - \frac{1}{4}} = \frac{3}{\frac{3}{4}} = 4 \).
Answer: Converges to 4.
Example 8: p-Series
Test \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \) for convergence.
- \( p = \frac{3}{2} > 1 \).
- Converges by p-series test.
Answer: Converges.
6) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Determine if the series converges or diverges using any appropriate test. 📚
\( \sum_{n=1}^{\infty} \frac{1}{n} \)
\( \sum_{n=1}^{\infty} \frac{1}{n^3} \)
\( \sum_{n=1}^{\infty} \frac{2^n}{n^2} \)
\( \sum_{n=1}^{\infty} \frac{1}{2^n} \)
\( \sum_{n=1}^{\infty} \frac{n}{n+1} \)
\( \sum_{n=1}^{\infty} \frac{1}{n(n+1)} \)
\( \sum_{n=1}^{\infty} \frac{3^n}{n!} \)
\( \sum_{n=2}^{\infty} \frac{1}{n \ln n} \)
\( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \)
\( \sum_{n=1}^{\infty} \frac{1}{(2n)!} \)
\( \sum_{n=1}^{\infty} \frac{n^2}{2^n} \)
Challenging Practice Questions 🌟
Instructions: Analyze the convergence of these series using multiple tests if needed. 🧠
Test \( \sum_{n=1}^{\infty} \frac{n^3}{2^n} \) using the ratio test and interpret the result.
Determine if \( \sum_{n=1}^{\infty} \frac{\sin(n)}{n^2} \) converges, considering oscillation.
Evaluate \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2} \) and identify its sum if it converges.
Compute the convergence of \( \sum_{n=2}^{\infty} \frac{1}{\ln(n) \cdot \ln(\ln(n))} \) using the integral test.
Analyze \( \sum_{n=1}^{\infty} \frac{n!}{n^n} \) using the ratio test and discuss its behavior.
7) Summary & Cheat Sheet 📋
7.1) Infinite Series
\( \sum a_n \), converges if \( \lim S_n = S \), diverges otherwise.
7.2) Convergence Tests
Ratio (\( L < 1 \)), Root (\( L < 1 \)), Integral (if \( \int f(x) \, dx \) converges).
7.3) Special Series
Geometric (\( |r| < 1 \)), p-Series (\( p > 1 \)).
You’ve mastered infinite series! Next, we’ll explore power series. 🎉