📐Analytic Geometry and Calculus Unit 12 – Infinite Sequences and Series

Key Concepts

• Sequences represent ordered lists of numbers that follow a specific pattern or rule
• Series are the sum of the terms in a sequence
• Convergence occurs when the sum of an infinite series approaches a finite value as the number of terms approaches infinity
• Divergence happens when the sum of an infinite series does not approach a finite value or grows without bound
• Tests for convergence include the ratio test, root test, integral test, and comparison tests which determine whether a series converges or diverges
• Sequences and series have applications in calculus such as approximating functions, evaluating improper integrals, and solving differential equations
  • Taylor and Maclaurin series expand functions as infinite series for approximation and analysis
• Understanding the behavior of sequences and series is crucial for advanced calculus concepts and problem-solving strategies

Definitions and Terminology

• Sequence: an ordered list of numbers, denoted as $a_n$ where $n$ represents the position of the term
• Series: the sum of the terms in a sequence, denoted as $\sum_{n=1}^{\infty} a_n$ for an infinite series
• Partial sum: the sum of the first $n$ terms of a series, denoted as $S_n = \sum_{i=1}^{n} a_i$
• Limit of a sequence: the value that the terms of a sequence approach as $n$ approaches infinity, denoted as $\lim_{n \to \infty} a_n$
• Convergence: a series converges if its sequence of partial sums approaches a finite limit
  • Sum of a convergent series: the limit of the sequence of partial sums, denoted as $S = \lim_{n \to \infty} S_n$
• Divergence: a series diverges if its sequence of partial sums does not approach a finite limit or grows without bound
• Absolute convergence: a series $\sum a_n$ converges absolutely if the series of absolute values $\sum |a_n|$ converges
• Conditional convergence: a series converges conditionally if it converges but not absolutely

Types of Sequences and Series

• Arithmetic sequence: a sequence where the difference between consecutive terms is constant, denoted as $a_n = a_1 + (n-1)d$
• Geometric sequence: a sequence where the ratio between consecutive terms is constant, denoted as $a_n = a_1 r^{n-1}$
• Harmonic series: the series $\sum_{n=1}^{\infty} \frac{1}{n}$, which diverges
• Alternating series: a series where the terms alternate in sign, such as $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}$
  • Leibniz test determines the convergence of alternating series
• p-series: the series $\sum_{n=1}^{\infty} \frac{1}{n^p}$, which converges for $p > 1$ and diverges for $p \leq 1$
• Power series: a series of the form $\sum_{n=0}^{\infty} a_n (x - c)^n$, where $c$ is the center of the series
  • Radius and interval of convergence determine where a power series converges

Convergence and Divergence

• Convergence tests determine whether a series converges or diverges
• Divergence test: if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges
• Comparison test: if $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ converges
  • Limit comparison test compares the limit of the ratio of two series
• Ratio test: if $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| < 1$, then the series $\sum a_n$ converges absolutely
• Root test: if $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$, then the series $\sum a_n$ converges absolutely
• Integral test: if $f(n) = a_n$ and $f(x)$ is continuous, positive, and decreasing on $[1, \infty)$, then $\sum a_n$ converges if and only if $\int_1^{\infty} f(x) dx$ converges
• Alternating series test: if $a_n$ is positive, decreasing, and $\lim_{n \to \infty} a_n = 0$, then the alternating series $\sum (-1)^{n+1} a_n$ converges

Tests for Convergence

• Ratio test: compares the limit of the ratio of consecutive terms
  • If $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$, then the series converges absolutely if $L < 1$, diverges if $L > 1$, and the test is inconclusive if $L = 1$
• Root test: compares the limit of the nth root of the absolute value of the terms
  • If $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L$, then the series converges absolutely if $L < 1$, diverges if $L > 1$, and the test is inconclusive if $L = 1$
• Integral test: compares the series to an improper integral
  • If $f(n) = a_n$, $f(x)$ is continuous, positive, and decreasing on $[1, \infty)$, then $\sum a_n$ converges if and only if $\int_1^{\infty} f(x) dx$ converges
• Comparison tests: compare the series to a known convergent or divergent series
  • Direct comparison test: if $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ converges; if $\sum b_n$ diverges, then $\sum a_n$ diverges
  • Limit comparison test: if $\lim_{n \to \infty} \frac{a_n}{b_n} = c > 0$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge
• Alternating series test: tests the convergence of alternating series
  • If $a_n$ is positive, decreasing, and $\lim_{n \to \infty} a_n = 0$, then the alternating series $\sum (-1)^{n+1} a_n$ converges

Applications in Calculus

• Power series representations of functions: express functions as power series for analysis and computation
  • Taylor series: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$, where $a$ is the center of the series and $f^{(n)}$ is the nth derivative of $f$
  • Maclaurin series: a Taylor series centered at $a = 0$
• Approximating functions: use partial sums of Taylor or Maclaurin series to approximate function values
• Evaluating improper integrals: use series representations to evaluate integrals with infinite limits or discontinuities
• Solving differential equations: use power series methods to find solutions to certain types of differential equations
• Fourier series: represent periodic functions as infinite sums of trigonometric functions
  • Applications in signal processing, heat transfer, and wave analysis

Common Pitfalls and Mistakes

• Misapplying convergence tests: using the wrong test or misinterpreting test results
• Confusing sequences and series: treating a sequence as a series or vice versa
• Mishandling absolute value: forgetting to consider absolute values when applying ratio or root tests
• Misidentifying the general term: incorrectly determining the formula for the nth term of a sequence or series
• Misusing the harmonic series: assuming the harmonic series converges or misapplying its properties
• Misinterpreting conditional convergence: assuming that conditional convergence implies absolute convergence
• Misapplying the alternating series test: using the test on non-alternating series or series that do not meet the test criteria
• Mishandling power series: incorrectly determining the radius or interval of convergence, or misapplying series operations

Practice Problems and Solutions

1. Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{n}{n^2 + 1}$
   • Solution: The series converges by the limit comparison test with $\sum \frac{1}{n}$
2. Find the radius and interval of convergence for the power series $\sum_{n=0}^{\infty} \frac{(x-2)^n}{3^n}$
   • Solution: The radius of convergence is $R = 3$, and the interval of convergence is $(2-3, 2+3) = (-1, 5)$
3. Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$ converges absolutely, conditionally, or diverges
   • Solution: The series converges absolutely by the alternating series test and the p-series test with $p = 2 > 1$
4. Approximate the value of $\sin(\frac{\pi}{3})$ using the first four terms of its Maclaurin series
   • Solution: $\sin(x) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}$, so $\sin(\frac{\pi}{3}) \approx 0.8660$
5. Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{2^n}{n!}$
   • Solution: The series converges by the ratio test with $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = \lim_{n \to \infty} \frac{2}{n+1} = 0 < 1$