🏃🏽‍♀️‍➡️Intro to Mathematical Analysis Unit 2 – Sequences and Limits in Mathematical Analysis

Key Concepts and Definitions

• Sequence: An ordered list of numbers, typically denoted as $\{a_n\}_{n=1}^{\infty}$ where $a_n$ represents the $n$-th term
• Index: The position or subscript of a term in a sequence (e.g., $n$ in $a_n$)
• Term: An individual number in a sequence, usually represented by a formula or expression involving the index
• Limit: The value that a sequence approaches as the index approaches infinity, denoted as $\lim_{n\to\infty} a_n = L$
  • If a limit exists, the sequence is said to converge; otherwise, it diverges
• Bounded sequence: A sequence where all terms lie between two fixed values (upper and lower bounds)
• Monotonic sequence: A sequence that is either non-increasing (each term is less than or equal to the previous term) or non-decreasing (each term is greater than or equal to the previous term)

Properties of Sequences

• Uniqueness: If a sequence converges, its limit is unique
• Boundedness: A convergent sequence is always bounded
  • The converse is not always true; a bounded sequence may not converge (oscillating sequences)
• Monotonicity: A monotonic and bounded sequence always converges
• Algebra of limits: If $\lim_{n\to\infty} a_n = L$ and $\lim_{n\to\infty} b_n = M$, then:
  • $\lim_{n\to\infty} (a_n + b_n) = L + M$
  • $\lim_{n\to\infty} (a_n - b_n) = L - M$
  • $\lim_{n\to\infty} (a_n \cdot b_n) = L \cdot M$
  • $\lim_{n\to\infty} (a_n / b_n) = L / M$ (provided $M \neq 0$)
• 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 $\lim_{n\to\infty} b_n = L$

Convergence and Divergence

• Convergence: A sequence $\{a_n\}$ converges to a limit $L$ if, for any $\varepsilon > 0$, there exists an $N$ such that $|a_n - L| < \varepsilon$ for all $n > N$
  • Intuitively, the terms of the sequence get arbitrarily close to the limit as $n$ increases
• Divergence: A sequence that does not converge to any finite limit is said to diverge
  • Divergence can occur if the sequence tends to infinity, negative infinity, or oscillates without settling on a specific value
• Cauchy sequence: A sequence $\{a_n\}$ is Cauchy if, for any $\varepsilon > 0$, there exists an $N$ such that $|a_n - a_m| < \varepsilon$ for all $n, m > N$
  • Every convergent sequence is Cauchy, and in complete metric spaces (like the real numbers), every Cauchy sequence converges
• Subsequence: A sequence formed by selecting a subset of terms from the original sequence while maintaining their order
  • If a sequence converges, then every subsequence converges to the same limit

Limit Theorems

• Limit of a constant sequence: If $a_n = c$ for all $n$, then $\lim_{n\to\infty} a_n = c$
• Limit of a multiple of a sequence: If $\lim_{n\to\infty} a_n = L$, then $\lim_{n\to\infty} (k \cdot a_n) = k \cdot L$ for any constant $k$
• Sum rule: If $\lim_{n\to\infty} a_n = L$ and $\lim_{n\to\infty} b_n = M$, then $\lim_{n\to\infty} (a_n + b_n) = L + M$
• Difference rule: If $\lim_{n\to\infty} a_n = L$ and $\lim_{n\to\infty} b_n = M$, then $\lim_{n\to\infty} (a_n - b_n) = L - M$
• Product rule: If $\lim_{n\to\infty} a_n = L$ and $\lim_{n\to\infty} b_n = M$, then $\lim_{n\to\infty} (a_n \cdot b_n) = L \cdot M$
• Quotient rule: If $\lim_{n\to\infty} a_n = L$ and $\lim_{n\to\infty} b_n = M$ with $M \neq 0$, then $\lim_{n\to\infty} (a_n / b_n) = L / M$
• Sandwich (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 $\lim_{n\to\infty} b_n = L$

Special Types of Sequences

• Arithmetic sequence: A sequence where the difference between consecutive terms is constant (common difference)
  • Example: $\{a_n\} = \{2, 5, 8, 11, 14, \ldots\}$ with common difference $d = 3$
• Geometric sequence: A sequence where the ratio between consecutive terms is constant (common ratio)
  • Example: $\{a_n\} = \{2, 6, 18, 54, 162, \ldots\}$ with common ratio $r = 3$
• Harmonic sequence: A sequence of the form $\{a_n\} = \{1/n\}_{n=1}^{\infty}$
  • The harmonic sequence converges to 0 but its series (the harmonic series) diverges
• Fibonacci sequence: A sequence where each term is the sum of the two preceding terms, starting with 0 and 1
  • Example: $\{a_n\} = \{0, 1, 1, 2, 3, 5, 8, 13, \ldots\}$
• Alternating sequence: A sequence where the signs of the terms alternate between positive and negative
  • Example: $\{a_n\} = \{1, -1, 1, -1, 1, -1, \ldots\}$

Techniques for Finding Limits

• Direct substitution: If the limit of a sequence can be evaluated by simply substituting the limiting value of $n$ into the expression for $a_n$, use this method
  • Example: $\lim_{n\to\infty} \frac{2n+1}{3n-4} = \frac{2}{3}$ (substitute $n=\infty$)
• Algebraic manipulation: Simplify the expression for $a_n$ using algebraic properties before attempting to find the limit
  • Example: $\lim_{n\to\infty} \frac{n^2+2n}{3n^2-1} = \lim_{n\to\infty} \frac{1+\frac{2}{n}}{3-\frac{1}{n^2}} = \frac{1}{3}$
• Squeeze theorem: If the sequence is bounded above and below by two sequences with the same limit, the original sequence converges to that limit
• Monotone convergence theorem: If a sequence is monotonic and bounded, it converges
  • To find the limit, evaluate the supremum (for non-decreasing) or infimum (for non-increasing) of the sequence
• Cauchy criterion: Prove that a sequence is Cauchy to show convergence without finding the limit explicitly

Applications in Analysis

• Continuity: A function $f$ is continuous at a point $a$ if $\lim_{x\to a} f(x) = f(a)$
  • Sequences can be used to prove or disprove continuity by considering the limit of $f(x_n)$ for any sequence $\{x_n\}$ converging to $a$
• Differentiability: A function $f$ is differentiable at a point $a$ if $\lim_{h\to 0} \frac{f(a+h)-f(a)}{h}$ exists
  • Sequences can be used to prove or disprove differentiability by considering the limit of the difference quotient for any sequence $\{h_n\}$ converging to 0
• Riemann integration: The Riemann integral of a function $f$ over an interval $[a,b]$ is defined as the limit of Riemann sums
  • Sequences of partitions and sample points are used to define and evaluate Riemann integrals
• Power series: A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n (x-c)^n$
  • The convergence of power series is determined by the limit of the terms $a_n (x-c)^n$ as $n$ approaches infinity

Common Pitfalls and Misconceptions

• Assuming that a bounded sequence always converges
  • Counterexample: The sequence $\{a_n\} = \{(-1)^n\}$ is bounded but oscillates between 1 and -1, thus diverging
• Misapplying limit theorems when the required conditions are not met
  • Example: Attempting to use the quotient rule when the limit of the denominator is 0
• Confusing the limit of a sequence with the limit of its series
  • Example: The harmonic sequence $\{1/n\}$ converges to 0, but the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges
• Incorrectly assuming that a sequence converges to a specific value without proof
  • Always use the definition of convergence or appropriate theorems to rigorously prove the limit
• Forgetting to consider the absolute value when using the definition of convergence
  • The condition $|a_n - L| < \varepsilon$ ensures that the terms get close to $L$ from both above and below
• Misinterpreting the index $n$ as a real number instead of a natural number
  • Sequences are defined for $n \in \mathbb{N}$, not for all real numbers