Lecture 2 – Improper Integrals Wrap-Up & Foundations of Sequences

Review of Improper Integrals

  • Goal: wrap-up last lecture’s unfinished examples (Examples 6–9) before starting Sequences.

  • Key technique revisited: Comparison Test for improper integrals.

• Example 9

  • Integral: \displaystyle\int_{1}^{\infty} e^{-x^2}\,dx

  • Compare with e^{-x} (easy antiderivative).

  • For x>1 we have e^{-x^2}\le e^{-x} (exponent more negative).

  • Since \int_{1}^{\infty} e^{-x}\,dx converges, the given integral also converges by Comparison Test.

• Positive Exponent Example

  • Integral: \displaystyle\int_{1}^{\infty} e^{x}\,dx

  • Write as \lim{t\to\infty}\int{1}^{t}e^{x}\,dx = \lim_{t\to\infty}(e^{t}-e^{1}) → \infty.

  • Therefore diverges (function grows rapidly).

• Trig / Rational Example

  • Integral: I=\displaystyle\int_{1}^{\infty}\frac{2+\sin x}{x^{2}}\,dx

  • Bound numerator: -1\le\sin x\le 1 \Rightarrow 1\le 2+\sin x\le 3.

  • Hence \dfrac{1}{x^{2}}\le \dfrac{2+\sin x}{x^{2}} \le \dfrac{3}{x^{2}}.

  • \int_{1}^{\infty} \dfrac{1}{x^{2}}\,dx converges (p–integral with p=2>1).

  • By Comparison Test, I converges.

    • Moral: keep Comparison Test in your analytic toolbox to avoid brute-force antiderivatives (e.g.", Gaussian integral" \int e^{-x^{2}}dx).

Introduction to Sequences

  • Sequence = ordered list of numbers indexed by positive integers; essentially a function a:\mathbb{N}\to\mathbb{R}.

  • Standard notation: (a{n}){n=1}^{\infty}\,,\;{a{n}}\,,\;\langle a{n}\rangle.

  • Graphically: plot only at integer n; looks like a function with “holes” elsewhere.

Examples

  1. a_{n}=\dfrac{n}{n+1}=\tfrac12,\tfrac23,\tfrac34,\dots

  2. Alternating sequence a_{n}=(-1)^{n}\dfrac{n+1}{3^{n}} (sign flip via (-1)^{n}).

    • Denominator 3^{n} dominates; magnitudes (\to0).

  3. Fibonacci (recursive)

    • F{1}=1,\;F{2}=1,\;F{n}=F{n-1}+F_{n-2} for n\ge3.

    • First five terms: 1,1,2,3,5.

    • Recursively defined sequences need not have a simple closed form.

Trend Concepts for Sequences
1. Monotonicity

  • Monotonic increasing: a{n}\le a{n+1}\,\,\forall n.

  • Strictly monotonic increasing: a{n}{n+1}\,\,\forall n (no equal repeats).

  • Analogous definitions for (strictly) monotonic decreasing.

  • "Monotonic" alone ⇒ either consistently non-decreasing or non-increasing.

2. Simple Moving Averages (SMA) – “Smoothing”

  • Fix a smoothing period k\in\mathbb{N}.

  • Original sequence: (a_{n}).

  • Smoothed sequence:
    b{n}=\dfrac{1}{k}\sum{j=n-k+1}^{n} a_{j}\,,\qquad n\ge k

  • Interpretation: mean of the last k observations; reduces noise, reveals underlying trend.

Illustrative Examples
• Baxter the German Shepherd

  • Data: grams of dog treats per week since birth.

  • Chose k=4; SMA (red) smoother & strictly increasing vs. raw weekly data (blue).

• Stock–Trading Heuristic

  • Price series (p_{n}) (e.g.", daily closes).

  • Short-term SMA: k_{s}=10.

  • Long-term SMA: k_{l}=50.

  • Trading rule:
    • If \text{SMA}{10} crosses     above \text{SMA}{50} ⇒ potential up-trend ⇒ buy.
    • If \text{SMA}{10} crosses     below \text{SMA}{50} ⇒ potential down-trend ⇒ sell.

• Study-Habits Analogy

  • Sequence: daily study hours before semester end.

  • Raw pattern: zeroes for weeks, spike pre-final ⇒ stressful.

  • Large-k SMA smooths to near-constant commitment; encourages balanced preparation.

Limits of Sequences: \varepsilon–N Definition

  • Sequence (a{n}) converges to L if: \forall\,\varepsilon>0\,\,\exists\,N\in\mathbb{N}\text{ s.t. } n>N\Rightarrow |a{n}-L|<\varepsilon. tric view: eventually (after index N) all terms lie within an \varepsilon-tube around L.

Quick Classification Examples

  1. a_{n}=\dfrac{1+2n}{n}=\dfrac1n+2 \;\to\;2.

  2. a{n}=\dfrac{4n^{2}+2n+1}{n^{3}-n^{2}+10} \Rightarrow factor n^{3} ⇒ a{n}=\dfrac{4+\tfrac{2}{n}+\tfrac{1}{n^{2}}}{n(1-\tfrac{1}{n}+\tfrac{10}{n^{3}})}\to0.

  3. a_{n}=2+\dfrac{\sin n}{n}

    • Bound: -\dfrac1n\le\dfrac{\sin n}{n}\le\dfrac1n; both extremes →0 ⇒ limit 2 (Squeeze Theorem).

  4. a_{n}=n diverges to +\infty.

  5. a_{n}=\cos\left(\dfrac{n\pi}{7}\right)

    • For n=7k, term =\cos(k\pi)=(-1)^{k} ⇒ infinitely many +1 and -1.

    • Cannot settle near a single L ⇒ diverges.

Limit of a Function: \varepsilon–\delta Definition

  • Statement: \displaystyle\lim_{x\to a}f(x)=b if
    \forall\,\varepsilon>0\,\exists\,\delta>0 \text{ s.t. } |x-a|<\delta \Rightarrow |f(x)-b|<\varepsilon.

  • Picture: choose \varepsilon-tube around b on y-axis; shrink \delta interval around a on x-axis until entire graph segment fits in tube.

  • Analogous "infinite limit" form: \displaystyle\lim_{x\to\infty}f(x)=L ⇔ \forall\varepsilon>0\,\exists M>0:\,x>M\Rightarrow |f(x)-L|<\varepsilon.

Continuity

  • Function f is continuous at l iff
    \displaystyle\lim_{x\to l}f(x)=f(l).

  • Limit Preservation Theorem (Plug-in Rule):
    If \displaystyle\lim{n\to\infty}a{n}=l and f is continuous at l, then
    \displaystyle\lim{n\to\infty}f(a{n})=f(l).

  • Visual: sequence points march toward l on x-axis; continuity carries their images toward f(l) on y-axis.

Squeeze (Sandwich) Theorem – Preview

  • For sequences (a{n}), (b{n}), (c{n}) with a{n}\le b{n}\le c{n}\,,\;\forall n\ge N{0} and \displaystyle\lim{n\to\infty}a{n}=\lim{n\to\infty}c{n}=L, then \displaystyle\lim{n\to\infty}b_{n}=L.

  • Same principle for functions.

  • Application already used above (Example 3) to handle 2+\frac{\sin n}{n}.

Take-Away Toolbox

  • Improper Integrals: Remember p–test & Comparison Test.

  • Sequences: identify through definition, monotonicity, recursion, smoothing (SMA), apply \varepsilon–N proofs if needed.

  • Limits/Continuity: internalize \varepsilon–\delta language; continuity lets you "plug-in" limits.

  • Squeeze Theorem: invaluable for oscillatory but diminishing terms.

Next lecture: deeper properties of sequences + full discussion of Squeeze Theorem.