Week 3a Study Notes: Functions, Sequences, and Series

Introduction to Functions and Problems

• Problem 1: Function Evaluation

  • Recall three fundamental tasks involving functions:

    • Evaluate: Given $x_0$, find $f(x_0)$.

    • Plot: Graph of $f(x)$ versus $x$.

    • Invert: Given $y_0$, find $x$ such that $y = f(x_0)$.

  • Evaluating functions for all real $x$:

    • How can we evaluate values such as $f(x) = x^{1/2}$, $f(x) = x^{2/3}$, or $f(x) = 10^x$?

    • Specific examples include finding \root \root 2 or $10^{\sqrt{2}}$.

• Problem 2: Function Plotting and Continuity

  • Questions raised regarding plotting:

    • How do we join discrete points to make a curve?

    • Why is a smooth curve justified in certain contexts?

    • What exactly defines "smooth"?

  • Physical variables often represented by smooth curves include:

    • Heat, temperature, and pressure.

    • Current and voltage.

    • Magnetic fields and electric fields.

  • Example graph mentioned: $y = |x|$.

• Problem 3: Growth, Fall, and Critical Points

  • Growth and Fall (Monotonicity):

    • Increasing: Occurs when x < x_1, x_2 < x < x_3, and x > x_4.

    • Decreasing: Occurs when x_1 < x < x_2 and x_3 < x < x_4.

  • Critical Points: A point $x \text{ in } \mathbb{R}$ where the graph of $f(x)$ attains a maximum or minimum value is called a critical point of $f$. At these points, the function hits a peak or trough and changes behavior.

  • Asymptotics:

    • For large, positive $x$: The function might tend toward a non-zero value.

    • For large, negative $x$: The function may tend toward zero.

Sequences: Definitions and Examples

• Definition of a Sequence:

  • A sequence of real numbers is a function from the set of non-negative integers $\{0, 1, 2, \dots\}$ to $\mathbb{R}$.

  • Notation: $f: \mathbb{N} \cup \{0\} \rightarrow \mathbb{R}$ or $f: \mathbb{N} \rightarrow \mathbb{R}$.

• Basic Examples:

  • $a_n = n$: Yields $0, 1, 2, 3, \dots$

  • $a_n = (-1)^n$: Yields $1, -1, 1, -1, \dots$

  • $a_n = \frac{1}{n+1}$: Yields $1, 1/2, 1/3, 1/4, \dots$

• Sequences Defined by Recursions:

  • Fibonacci Sequence:

    • Defined by $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = a_n + a_{n-1}$ for $n \ge 1$.

    • Terms: $0, 1, 1, 2, 3, 5, 8, 13, 21, \dots$

  • Arithmetic Progression (A.P.):

    • Defined by recursive relation: $a_{n+1} = a_n + d$ for $n \ge 0$, where $d$ is the common difference.

    • General term: $a_n = a_0 + nd$.

    • Example provided: $a_0 = 2$, $d = -1$. Terms: $2, 1, 0, -1, -2$. Calculation for $a_4 = a_0 + 4d = 2 - 4 = -2$.

  • Geometric Progression (G.P.):

    • Defined by recursive relation: $a_{n+1} = a_n r$ for $n \ge 0$, where $r$ is the common ratio.

    • General term: $a_n = a_0 r^n$.

  • $\sqrt{2}$ Sequence (Approximation):

    • Recursion: $a_0 = 2$, $a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}$ for $n \ge 0$.

    • Terms include: $a_0 = 2$, $a_1 = 1.5$, $a_2 = 1.416666\dots$, $a_3 = \frac{577}{408} ≈ 1.414215686\dots$

    • Note: While $\sqrt{2}$ is irrational ($\sqrt{2} \ne p/q$), the terms in this sequence are rational and can be used to approximate its value ($1.4142135623730951\dots$) as the sequence flattens and tends toward it.

• Visualizing Sequences:

  • Sequences are typically visualized using stem plots where $a_n$ is plotted against the index $n$.

  • Specific plot behaviors identified:

    • $a_n = n$ (Increasing growth).

    • $a_n = (-1)^n$ (Alternating between $1$ and $-1$).

    • $a_n = \frac{1}{n+1}$ (Decreasing toward zero).

    • $a_n = \frac{2 + 5n}{9}$ (Increasing).

    • $a_n = 3 \times 2^n$ (Increasing).

    • $a_n = 3 \times 0.5^n$ (Decreasing).

Series: Sequences Defined as Summations

• Definition of a Series:

  • Given a sequence $\{a_n\}$ for $n = 1, 2, \dots$, we define a "summation" sequence $s_n$ (the series).

  • Recursive definition: $s_1 = a_1$, $s_{n+1} = s_n + a_{n+1}$ for $n = 1, 2, \dots$

  • Explicit definition: $s_n = a_1 + a_2 + \dots + a_n = ∑_{k=1}^n a_k$.

• Examples of Series:

  • Sum of Natural Numbers:

    • $a_n = n$.

    • $s_n = 1 + 2 + \dots + n = ∑_{k=1}^n k = \frac{n(n+1)}{2}$.

    • Partial sums: $1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$

  • Harmonic Series ($H_n$):

    • $H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} = ∑_{k=1}^n \frac{1}{k}$.

  • Harmonic Series of Order 2 ($H_n^{(2)}$):

    • $H_n^{(2)} = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} = ∑_{k=1}^n \frac{1}{k^2}$.

  • Alternating Harmonic Series:

    • $a_n = \frac{(-1)^{n-1}}{n}$.

    • $s_n = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots + \frac{(-1)^{n-1}}{n} = ∑_{k=1}^n \frac{(-1)^{k-1}}{k}$.

  • Madhava-Leibniz $\pi/4$ Series:

    • $a_n = \frac{(-1)^{n-1}}{2n-1}$.

    • $s_n = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots + \frac{(-1)^{n-1}}{2n-1} = ∑_{k=1}^n \frac{(-1)^{k-1}}{2k-1}$.

Properties of Sequences: Boundedness and Monotonicity

• Boundedness Definitions:

  • Bounded Above: A sequence $\{a_n\}$ is bounded above (upper-bounded) by $M$ if $a_n \le M$ for all $n$.

  • Bounded Below: A sequence $\{a_n\}$ is bounded below (lower-bounded) by $m$ if $a_n \ge m$ for all $n$.

• Examples of Boundedness:

  • $a_n = n$: Lower bound = $0$, Upper bound = None.

  • $a_n = (-1)^n$: Lower bound = $-1$, Upper bound = $1$.

  • $a_n = \frac{1}{n+1}$: Lower bound = $0$, Upper bound = $1$.

  • $a_n = -2^n$: Lower bound = None, Upper bound = $-1$.

  • $a_n = (-2)^n$: Lower bound = None, Upper bound = None.

• Boundedness in Special Sequences:

  • Arithmetic Progression ($a_n = a_0 + nd$):

    • If d > 0: Bounded below by $a_0$, not bounded above.

    • If d < 0: Bounded above (at $a_0$), not bounded below.

  • Geometric Progression ($a_n = 2r^n$):

    • If r > 1: Lower bound = $2$, Upper bound = None.

    • If 0 < r < 1: Lower bound = $0$, Upper bound = $2$.

  • $\sqrt{2}$ Sequence: Lower bound = $1$, Upper bound = $2$.

• Monotonicity Definitions:

  • Increasing: a_{n+1} > a_n for all $n$.

  • Non-decreasing: $a_{n+1} \ge a_n$ for all $n$.

  • Decreasing: a_{n+1} < a_n for all $n$.

  • Non-increasing: $a_{n+1} \le a_n$ for all $n$.

• Examples of Monotonicity:

  • $a_n = n$: Increasing.

  • $a_n = (-1)^n$: None.

  • $a_n = 1/n$: Decreasing.

  • $a_n = -2^n$: Decreasing.

  • $a_n = (-2)^n$: None.

  • A.P. (d > 0): Increasing.

  • A.P. (d < 0): Decreasing.

  • G.P. (r > 1): Increasing.

  • G.P. (0 < r < 1): Decreasing.

  • $\sqrt{2}$ Sequence: Decreasing (after $a_0$).

Limit of a Sequence

• Definition of Limit:

  • A sequence $\{a_n\}$ tends to a limit $L$ if for any \epsilon > 0, we can find an integer $N$ such that $L - \epsilon \le a_n \le L + \epsilon$ for all n > N_{\epsilon}.

  • Notation: $\lim_{n \rightarrow \infty} a_n = L$.

• Classifying Sequences Based on Limits:

  • Converges: If it has a limit that is finite.

  • Diverges to $\infty$: If it grows unbounded and is positive.

  • **Diverges to $-\infty:** If it grows unbounded and is negative.\n * **No Limit:** If none of the above are true.\n * Note: \infty denotes that the value keeps growing infinitely.\n\n* **Limit Examples (n = 1, 2, \dots):**\n * a_n = 1$: Limit as$n \rightarrow \infty$is$1.\n * a_n = 1/n$: Limit as$n \rightarrow \infty$is$0.\n * a_n = n$: Diverges toward$\infty.\n * a_n = -2^n$: Diverges toward$-\infty.\n * a_n = (-1)^n$: None (Sequence fluctuates between$1$and$-1).\n * a_n = (-2)^n: None.\n\n# Questions & Discussion\n\n* **Question (Sequence Identification):** Which of the following functions represents a sequence?\n * a) \mathbb{R} \rightarrow \mathbb{R}\n * b) f_2: \mathbb{N} \cup {0} \rightarrow \mathbb{R}\n * c) f_3: \mathbb{R}^{-} \rightarrow \mathbb{R}\n * d) f_4: \mathbb{R}^{+} \rightarrow \mathbb{R}\n * e) f_5: \mathbb{N} \cup {0} \rightarrow \mathbb{Z}\n * *Note: According to the definition, a sequence is a function from non-negative integers to real numbers, making (b) and (e) valid candidates as \mathbb{Z} \subset \mathbb{R}.*\n\n* **Question (Arithmetic Progression):** What is the common difference in the arithmetic progression 5, 14, 23, 32, 41, 50, \dots?\n * Calculation: 14 - 5 = 9$. The common difference is$9.\n\n* **Question (Recursion Formula):** Consider the sequence based on the continued fraction structure: a_0 = 2$,$a_1 = 2 - 1/2 = 3/2$,$a_2 = 2 + 1/(2-1/2), etc. Identify the correct recursion formula.\n\n* **Question (Series Summation):** Consider a series S = ∑_{n=1}^{\infty} rac{1}{(2n+1)3^n}$. Which option correctly identifies the partial sum$ s_n?\n * Correct Option: s_n = rac{1}{3 \cdot 3} + rac{1}{5 \cdot 3^2} + \frac{1}{7 \cdot 3^3} + \dots + rac{1}{(2n+1)3^n}.\n\n* **Question (Monotonicity Check):** Identify correct statements regarding increasing/decreasing sequences:\n * Is the sequence {1 - 1/n^2} decreasing? (No, it is increasing).\n * Is the sequence {-1/n} an increasing sequence? (Yes).\n * Is the sequence {n / (n+1)} an increasing sequence? (Yes).\n * Is the sequence {(4/3)^n}$$ an increasing sequence? (Yes).