Lecture 3 Notes – Sequences, ε–δ Limits, and Introduction to Series

Administrative Announcements

• New class representatives elected: Isabella Alcznis, Leila Brewster, Mike Locke

  • Role: liaison between students and teaching/admin staff for feedback or concerns.

• Workshops begin this week

  • Each student should know their assigned room/demonstrator.

  • 30-min post-workshop consultations function as extra office hours.

• All logistical details (workshop rosters, notes, consultation times) posted on the course website.

Review of ε–δ Limits (Sequences → Functions)

Definition & Visual Intuition

• To prove $\lim_{x\to a}f(x)=b$ we must show:
  \forall\,\varepsilon>0\;\exists\,\delta(\varepsilon)>0:\;0<|x-a|<\delta\;\Rightarrow\;|f(x)-b|<\varepsilon

• Geometric picture:

  • Draw a horizontal “tube” of height $\varepsilon$ around $y=b$.

  • Find a vertical window of width $\delta$ around $x=a$ that forces the graph inside that tube.

• Think of a game: “ε-player” picks ε, “δ-player” must respond with a δ functionally dependent on ε.

Worked Example 1: Point-wise Limit

Prove $\displaystyle\lim_{x\to1}(2x^2+x)=3$.

1. Want |2x^2+x-3|<\varepsilon whenever |x-1|<\delta.

2. Factor: $2x^2+x-3=(2x+3)(x-1).$

3. Pull out $|x-1|$: $|2x^2+x-3|\le|2x+3|\,|x-1|.$

4. Force $|2x+3|$ to be small by choosing $\delta\le1$ so |x|<2 and |2x+3|<7.

5. Obtain |2x^2+x-3|<7\delta.

6. Make 7\delta<\varepsilon → pick $\boxed{\delta=\varepsilon/8}$ (any <\varepsilon/7 works).

Worked Example 2: Limit at Infinity

Goal: $\displaystyle\lim_{x\to\infty}\frac{\sin x}{x^2+2}=0.$

1. Need \left|\dfrac{\sin x}{x^2+2}\right|<\varepsilon for large $x$.

2. Use bounds: $|\sin x|\le1$ and x^2+2>x^2 ⇒ |\sin x|/(x^2+2)<1/x^2.

3. Require 1/x^2<\varepsilon → x>1/\sqrt{\varepsilon}.

4. Choose $\boxed{N(\varepsilon)=\lceil1/\sqrt{\varepsilon}\rceil+1}$; then x>N guarantees inequality.

Squeeze (Sandwich) Theorem for Sequences

• If $a<em>n\le b</em>n\le c<em>n$ for all $n$ and $\lim a</em>n=\lim c<em>n=L$, then $\lim b</em>n=L.$

• Visual: two “bookend” sequences converge to the same height, forcing the middle one along.

Bounding Examples

1. $b_n=\dfrac1{n^2-2n}$

   • Dominant term $n^2$; construct $a<em>n=\dfrac1{1000n^2},\;c</em>n=\dfrac{1000}{n^2}.$

   • Both $a<em>n,c</em>n\to0$ ⇒ $b_n\to0.$

2. $b<em>n=\dfrac1{n^3+2n}$ → use $1/1000n^3\le b</em>n\le1000/n^3$ ⇒ $b_n\to0.$

3. $b_n=\dfrac{\sin n}{n^2}$ → since $|\sin n|\le1$, squeeze by $\pm1/n^2$ ⇒ limit 0.

Exercise: Rigorously justify each inequality (choose a concrete index beyond which it holds).

Boundedness & Monotone Convergence

Definitions

• Bounded above: $\exists M:\;a_n\le M$ ∀n.

• Bounded below: $\exists m:\;a_n\ge m$ ∀n.

• Bounded: both above and below.

• Monotone: sequence is either non-decreasing (monotone ↑) or non-increasing (monotone ↓).

Theorem – Monotone Convergence

Every bounded monotonic sequence converges.

• For monotone ↑: limit is $\displaystyle\sup<em>{n\in\Bbb N}a</em>n.$

• For monotone ↓: limit is $\displaystyle\inf<em>{n\in\Bbb N}a</em>n.$

Suggested Proof Exercises

1. Prove: “Every convergent sequence is bounded.”

2. Provide a bounded sequence that does not converge (e.g.
   $(-1)^n$).

Series = Infinite Sums

Formal Definition via Partial Sums

• Given a sequence ${a<em>n}_{n=1}^\infty$, define partial sums $S</em>N=\sum<em>{n=1}^{N}a</em>n.$

• The series $\sum<em>{n=1}^{\infty}a</em>n$ converges to $L$ if $\displaystyle\lim<em>{N\to\infty}S</em>N=L.$

Divergent Example – Grandi’s Series

• Series $1-1+1-1+\dots$ has partial sums $1,0,1,0,\dots$ which do not converge ⇒ series diverges.

Telescoping Series

• Example: $\sum_{n=1}^{\infty}\left(\frac1n-\frac1{n+1}\right)$.

  • Partial sum $S_N=1-\dfrac1{N+1}.$

  • Limit $\displaystyle\lim<em>{N\to\infty}S</em>N=1$ ⇒ convergent.

Geometric Series

General Form & Sum Formula

• $\displaystyle\sum_{n=0}^{\infty}ar^{n}=a+ar+ar^{2}+\dots$

• Partial sum derivation (multiply by $1-r$ and telescope):
  $S_N=\frac{a-a r^{N+1}}{1-r}\qquad (r\ne1).$

Convergence Test

• Convergent iff |r|<1 with limit $\displaystyle S=\frac{a}{1-r}.$

• Divergent if $|r|\ge1$ (unless $a=0$).

Sample Calculations

1. $\sum_{n=1}^{\infty}e^{-n}$

   • Rewrite as $a=e^{-1},\;r=e^{-1}$ ⇒ |r|<1.

   • Sum $= \dfrac{e^{-1}}{1-e^{-1}}.$

2. $\sum_{n=0}^{\infty}\dfrac{3}{\pi^{n+1}}$

   • $a=3/\pi,\;r=1/\pi$ ⇒ convergent; sum $=\dfrac{3/\pi}{1-1/\pi}.$

3. (Preview) Find smallest p>0 making $\sum_{n=0}^{\infty}(1+p)^{n}$ divergent → use $|r|\ge1$ criterion.

Necessary Condition & Divergence Test

• If $\sum a<em>n$ converges, then necessarily $\displaystyle\lim</em>{n\to\infty}a_n=0.$

• Contrapositive (Divergence Test): If $\lim a_n\neq0$ or does not exist, the series diverges.

Example

• $\sum_{n=1}^{\infty}\sin n$ diverges because $\sin n$ does not approach 0 (values oscillate on the unit circle).

Algebraic Laws for Series

For two convergent series $\sum a<em>n,\;\sum b</em>n$ and constant $c$:

1. $\sum c a<em>n$ converges (to $c\sum a</em>n$).

2. $\sum (a<em>n\pm b</em>n)=\sum a<em>n\pm\sum b</em>n$ is convergent.

3. Any finite linear combination of convergent series is convergent.

• Justification: apply the corresponding limit rules to the sequence of partial sums.

Exercises & Self-Study Prompts

• Work additional ε–δ proofs from the textbook.

• Construct explicit bounds to complete the squeeze-theorem inequalities posed in class.

• Prove monotone convergence theorem rigorously.

• Show boundedness ≠ convergence by giving at least one concrete counter-example.

• Re-derive the geometric-series formula using the telescoping trick.

• Use the divergence test on exotic series (e.g. $\sum \cos(\sqrt n)$).

Looking Ahead

• Next lecture:

  • Finish geometric-series parameter problem.

  • Introduce additional convergence/divergence tests (Integral, Comparison, Ratio, etc.).

• Keep using workshop consultations to clarify proofs; epsilon-delta manipulations and bounding tricks will appear throughout the series unit.