🏃🏽‍♀️‍➡️Intro to Mathematical Analysis Unit 11 – Sequences and Series in Function Analysis

Key Concepts and Definitions

• Sequence: An ordered list of numbers, denoted as $\{a_n\}_{n=1}^{\infty}$ where $a_n$ represents the $n$-th term
  • Example: $\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...\}$ is a sequence where $a_n = \frac{1}{n}$
• Series: The sum of the terms in a sequence, denoted as $\sum_{n=1}^{\infty} a_n$
  • Example: $\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + ...$
• Partial sum: The sum of the first $n$ terms of a series, denoted as $S_n = \sum_{k=1}^{n} a_k$
• Convergence: A sequence or series approaches a finite limit as $n$ approaches infinity
  • Example: The sequence $\{\frac{1}{n}\}_{n=1}^{\infty}$ converges to 0
• Divergence: A sequence or series does not approach a finite limit as $n$ approaches infinity
  • Example: The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges
• Cauchy sequence: A sequence $\{a_n\}$ where for any $\varepsilon > 0$, there exists an $N$ such that for all $m, n > N$, $|a_m - a_n| < \varepsilon$
• Monotone sequence: A sequence that is either non-increasing or non-decreasing

Types of Sequences and Series

• Arithmetic sequence: A sequence where the difference between consecutive terms is constant
  • Example: $\{2, 5, 8, 11, ...\}$ is an arithmetic sequence with common difference 3
• Geometric sequence: A sequence where the ratio between consecutive terms is constant
  • Example: $\{2, 6, 18, 54, ...\}$ is a geometric sequence with common ratio 3
• Harmonic sequence: A sequence of the form $\{\frac{1}{n}\}_{n=1}^{\infty}$
• Alternating series: A series where the terms alternate in sign
  • Example: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ...$
• Power series: A series of the form $\sum_{n=0}^{\infty} a_n(x - c)^n$, where $c$ is the center of the series
  • Example: The Maclaurin series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$
• Taylor series: A power series that approximates a function near a specific point
• Fourier series: A series representing a periodic function as a sum of sines and cosines

Convergence and Divergence

• Absolute convergence: A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges
  • Implies convergence, but not vice versa
• Conditional convergence: A series that converges, but not absolutely
  • Example: The alternating harmonic series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges conditionally
• Radius of convergence: The radius $R$ of the largest interval $(-R, R)$ on which a power series converges
• Interval of convergence: The interval on which a power series converges
  • Determined by the radius of convergence and the behavior at the endpoints
• Pointwise convergence: A sequence of functions $\{f_n\}$ converges pointwise to $f$ if for each $x$ in the domain, $\lim_{n \to \infty} f_n(x) = f(x)$
• Uniform convergence: A stronger form of convergence where the rate of convergence is independent of $x$
  • Implies pointwise convergence, but not vice versa

Limit Properties and Theorems

• Limit laws: Properties that allow for the manipulation of limits, such as the sum, difference, product, and quotient rules
  • Example: If $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$, then $\lim_{n \to \infty} (a_n + b_n) = A + B$
• Squeeze theorem: If $a_n \leq b_n \leq c_n$ for all $n$ and $\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$
  • Useful for finding limits of sequences bounded by two other sequences with known limits
• Monotone convergence theorem: A bounded monotone sequence converges
  • Example: The sequence $\{1 - \frac{1}{n}\}_{n=1}^{\infty}$ is increasing and bounded above by 1, so it converges
• Cauchy criterion: A sequence $\{a_n\}$ converges if and only if it is a Cauchy sequence
• Comparison test: Compares a series to another series with a known convergence behavior
  • If $\sum a_n$ and $\sum b_n$ are series with non-negative terms and $a_n \leq b_n$ for all $n$, then:
    • If $\sum b_n$ converges, then $\sum a_n$ converges
    • If $\sum a_n$ diverges, then $\sum b_n$ diverges

Tests for Convergence

• Ratio test: For a series $\sum a_n$, if $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$, then:
  • If $L < 1$, the series converges absolutely
  • If $L > 1$, the series diverges
  • If $L = 1$, the test is inconclusive
• Root test: For a series $\sum a_n$, if $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L$, then:
  • If $L < 1$, the series converges absolutely
  • If $L > 1$, the series diverges
  • If $L = 1$, the test is inconclusive
• Integral test: Compares a series to an improper integral
  • If $f(n) = a_n$ for a positive, decreasing function $f$, then $\sum a_n$ converges if and only if $\int_1^{\infty} f(x) dx$ converges
• Alternating series test: For an alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} b_n$, if $\{b_n\}$ is decreasing, non-negative, and $\lim_{n \to \infty} b_n = 0$, then the series converges
• Limit comparison test: Compares the limit of the ratio of corresponding terms of two series
  • If $\sum a_n$ and $\sum b_n$ are series with positive terms and $\lim_{n \to \infty} \frac{a_n}{b_n} = c > 0$, then either both series converge or both series diverge

Operations on Sequences and Series

• Addition and subtraction: If $\{a_n\}$ and $\{b_n\}$ are convergent sequences, then $\{a_n \pm b_n\}$ is also convergent
  • $\lim_{n \to \infty} (a_n \pm b_n) = \lim_{n \to \infty} a_n \pm \lim_{n \to \infty} b_n$
• Multiplication by a constant: If $\{a_n\}$ is a convergent sequence and $c$ is a constant, then $\{ca_n\}$ is also convergent
  • $\lim_{n \to \infty} (ca_n) = c \cdot \lim_{n \to \infty} a_n$
• Product of sequences: If $\{a_n\}$ and $\{b_n\}$ are convergent sequences, then $\{a_n \cdot b_n\}$ is also convergent
  • $\lim_{n \to \infty} (a_n \cdot b_n) = \lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n$
• Cauchy product: The product of two series $\sum a_n$ and $\sum b_n$ is defined as $\sum c_n$, where $c_n = \sum_{k=0}^{n} a_k b_{n-k}$
  • If both series converge absolutely, then their Cauchy product also converges to the product of their sums
• Term-by-term differentiation and integration: If a power series converges uniformly, then it can be differentiated or integrated term-by-term within its interval of convergence
  • The resulting series will have the same radius of convergence as the original series

Applications in Function Analysis

• Approximating functions: Taylor and Maclaurin series can be used to approximate functions near a specific point
  • Example: The Maclaurin series for $\sin(x)$ is $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$
• Solving differential equations: Power series can be used to solve certain types of differential equations
  • Example: The power series solution to the differential equation $y'' + y = 0$ is $y = c_1 \cos(x) + c_2 \sin(x)$
• Evaluating improper integrals: Series can be used to evaluate improper integrals that are difficult to compute directly
  • Example: $\int_0^1 \frac{\ln(x)}{1-x} dx = \sum_{n=1}^{\infty} \frac{1}{n^2}$
• Representing periodic functions: Fourier series can be used to represent periodic functions as a sum of sines and cosines
  • Useful in signal processing and wave analysis
• Proving identities: Series manipulations can be used to prove mathematical identities
  • Example: The Basel problem, $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$, can be proved using Fourier series

Common Pitfalls and Tips

• Misusing tests for convergence: Be careful when applying tests for convergence, as each test has its own limitations and requirements
  • Example: The ratio test is inconclusive when the limit of the ratio is 1
• Forgetting to check endpoints: When determining the interval of convergence for a power series, don't forget to check the behavior at the endpoints
  • The series may converge conditionally, absolutely, or diverge at the endpoints
• Confusing sequences and series: Remember that a sequence is an ordered list of numbers, while a series is the sum of the terms in a sequence
  • Convergence of a sequence does not imply convergence of the corresponding series, and vice versa
• Mishandling conditional convergence: When a series converges conditionally, rearranging the terms can change the sum or cause the series to diverge
  • Example: The alternating harmonic series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges to $\ln(2)$, but rearranging the terms can make it converge to any real number or diverge
• Overrelying on intuition: Intuition can be misleading when dealing with sequences and series, especially when it comes to convergence and divergence
  • Always use formal definitions and tests to verify your intuition
• Practicing regularly: The best way to master sequences and series is through regular practice and exposure to various problems
  • Work through problems from textbooks, past exams, and online resources to reinforce your understanding