Calculus I · Patterns That Go Somewhere

Where this is heading

The word “approaches” is the hinge of everything that follows. In Chapter 2 we ask what a function approaches as its input closes in on a point — the same idea, lifted from lists to curves. In Chapter 3 we use it to do the thing this whole book is named for: take the slope of a straight line between two points on a curve and watch what it approaches as the points slide together. That limit is the slope of the curve itself — the derivative. Every limit you compute now is practice for that moment.

And $e$, which appeared here as nothing more than the destination of a climbing list, will turn out to be one of the two or three most important numbers in all of mathematics. You met it first as a sequence. That’s fitting: in this book, the deep things arrive quietly, as patterns that go somewhere.

Problems

Six tiers, each with a different job. Fluency and Practice are doing* — drill until the moves are automatic, then multi-step work where you pick the tool. Concept & counterexample and ◆ Diamond are thinking — traps, constructions, and Layer-2 proofs that test whether you understand, not just compute. Teach it is teaching — explain, write up, or fix reasoning for a peer, which is the surest way to find the gaps in your own. Looking ahead hands you Chapter 2 early. Do all of Fluency; do not skip Concept or Teach it — that is where the understanding actually sets.*

Fluency — do these fast, until they’re automatic

1. First four terms of: (a) $a_n=3n-1$; (b) $a_n=\left(\tfrac12\right)^n$; (c) $a_n=\dfrac{n}{n+1}$; (d) $a_n=\dfrac{(-1)^n}{n}$.
2. Arithmetic, geometric, or neither — and give $d$ or $r$ where it applies: (a) $7,10,13,16,\ldots$; (b) $3,-6,12,-24,\ldots$; (c) $1,\tfrac12,\tfrac13,\tfrac14,\ldots$; (d) $2,5,10,17,\ldots$.
3. Use a formula, not a list: (a) the $15$th term of the arithmetic sequence with $a_1=4,\ d=5$; (b) the $6$th term of the geometric sequence with $a_1=2,\ r=3$.
4. Guess the limit (or say “no limit”): (a) $\tfrac1n$; (b) $\tfrac{2n-1}{n}$; (c) $(-1)^n$; (d) $(0.8)^n$; (e) $n^2$; (f) $\tfrac{(-1)^n}{n}$.
5. Divide by the highest power: (a) $\displaystyle\lim\frac{3n+1}{n+2}$; (b) $\displaystyle\lim\frac{n^2-1}{2n^2+n}$; (c) $\displaystyle\lim\frac{5n}{n^2+1}$; (d) $\displaystyle\lim\frac{2n^3}{n^2+1}$.
6. For each $r$, does $r^n\to0$, $\to1$, blow up, or oscillate? $r=\tfrac13$; $r=-\tfrac12$; $r=1$; $r=-1$; $r=2$.

Practice — choose the right tool

1. $\displaystyle\lim_{n\to\infty}\frac{n^2-3n}{2n^2+1}$.
2. An $\infty-\infty$ that hides a finite answer: $\displaystyle\lim_{n\to\infty}\left(\sqrt{n^2+n}-n\right)$. (Rationalize, then divide by the top power.)
3. An $\infty\cdot0$: $\displaystyle\lim_{n\to\infty} n\left(\sqrt{1+\tfrac1n}-1\right)$. (Mirror Fight one from the chapter exactly.)
4. The $e$-family: (a) $\displaystyle\lim\left(1+\frac3n\right)^n$; (b) $\displaystyle\lim\left(1-\frac2n\right)^n$; (c) $\displaystyle\lim\left(1+\frac1n\right)^{3n}$.
5. Use the gauge “$(\text{exponent})\times(\text{base}-1)\to k \Rightarrow$ limit $=e^k$”: evaluate or classify $\left(1+\frac1n\right)^{n^2}$, $\left(1+\frac{1}{n^2}\right)^{n}$, and $\left(1+\frac5n\right)^{n}$. State $k$ first each time.
6. A self-defined sequence: $a_1=1$, $a_{n+1}=\sqrt{12+a_n}$. Compute $a_2,a_3$, then find the limit by letting $L$ satisfy the rule. (You’ll prove it converges in the Diamond tier.)
7. A geometric sequence has $a_1=1,\ r=0.9$. About how many terms until it drops below $0.001$? (Use the idea behind the $r^n\to0$ proof, or trial and error.)
8. Decompose, then take the limit: $\displaystyle\lim_{n\to\infty}\frac{n+(-1)^n}{n}$. (Split into two pieces, one obvious, one you’ve already handled.)

Concept & counterexample — think; almost no arithmetic. State true/false and justify, or build the example asked for.

1. 1. If $a_n\to L$, then every term is within $0.01$ of $L$. (b) If a sequence is bounded, it converges. (c) If $a_n\to0$, the sequence is eventually decreasing. (d) A sequence can converge to two different limits.
2. Construct a bounded sequence that does not converge.
3. Construct an increasing sequence that does not converge — and explain in one line which hypothesis of the Monotone Convergence principle your example is missing.
4. Construct a sequence that converges to $5$ but whose terms are sometimes above and sometimes below $5$.
5. Find the error. A student writes: “$(-1)^n$ gets arbitrarily close to $1$ infinitely often and to $-1$ infinitely often, so it has two limits, $1$ and $-1$.” Identify precisely what’s wrong, and state what is true.
6. Is it possible? A sequence with $a_n\to0$ whose running totals $a_1+a_2+\cdots+a_n$ nonetheless grow without bound. (Name an example if yes — this is a quiet preview of infinite series.)
7. The Monotone Convergence principle needs two hypotheses, “increasing” and “bounded above.” Give one sequence showing the conclusion fails if you drop “bounded above,” and one showing it fails if you drop “increasing.” (This proves both hypotheses are load-bearing.)

◆ Diamond — Layer 2: prove it

1. ◆ Prove $\dfrac1n\to0$ from the $\varepsilon$–$N$ definition.
2. ◆ Prove $\dfrac{1}{n^2}\to0$ from the definition.
3. ◆ Prove $\dfrac{n}{n+1}\to1$ from the definition. (First show $\left|\tfrac{n}{n+1}-1\right|=\tfrac{1}{n+1}$.)
4. ◆ Prove $\dfrac{2n+1}{n}\to2$ from the definition.
5. ◆ Suppose $a_n\to5$. Using the limit laws, find $\displaystyle\lim\left(3a_n-\tfrac1n\right)$, naming the law at each step.
6. ◆ Prove $\dfrac{\sin n}{n}\to0$ via the Squeeze Theorem: name the two walls, cite the theorem, and explain in one sentence why $\sin n$ never needed a limit of its own.
7. ◆ Prove the uniqueness of limits: a sequence cannot converge to two different values $L\ne M$. (Hint: if it did, take $\varepsilon=\tfrac{|L-M|}{3}$ and derive a contradiction — a term can’t be within $\varepsilon$ of both.) Connect this to your answer in 15(d).
8. ◆ For the sequence in problem 12, prove it is increasing and bounded above by $4$, completing the argument that it converges.
9. ◆ Prove Bernoulli’s inequality $(1+h)^n\ge1+nh$ for $h>-1$, $n\ge1$, by induction.
10. ◆◆ Prove $\left(1+\tfrac1n\right)^n<3$ for every $n$. Steps: expand by the binomial theorem; bound the $k$-th term by $\tfrac{1}{k!}$; bound $\tfrac{1}{k!}\le\tfrac{1}{2^{k-1}}$; sum the geometric bound to get below $1+\left(1+\tfrac12+\tfrac14+\cdots\right)=3$.
11. ◆◆ Adapt the $\sqrt[n]{n}\to1$ proof to show $\sqrt[n]{2^n+3^n}\to3$. (Factor out the dominant $3^n$, then squeeze.)

Teach it — the surest test of mastery is explaining it

1. A friend has only ever heard “a limit is what a sequence gets closer and closer to.” In three or four sentences written to them, explain why that phrasing is unsafe, using a single concrete sequence that “gets closer” for a while but does not converge.
2. Write a complete, textbook-quality worked solution to $\displaystyle\lim_{n\to\infty}\frac{3n^2+2n}{n^2-5}$, narrating each step in words a classmate could follow without you in the room.
3. Debug your classmate. They argue: “$\left(1+\tfrac1n\right)^n\to1$, because $1+\tfrac1n\to1$ and $1$ to any power is $1$.” Pinpoint the exact flawed step and write the one-sentence correction. (Name the indeterminate form they walked into.)
4. Design a problem. Invent a self-defined sequence $a_{n+1}=\sqrt{c+a_n}$ whose limit is exactly $5$. State the $c$ you need, then write the four-step solution key (bounded, increasing, converges, solve for $L$) as if for a teacher’s answer book.
5. In two or three sentences, teach the difference between “bounded” and “convergent.” Give one sequence that is bounded but not convergent, and say in plain words why one property does not imply the other.

Looking ahead — Chapter 2 in your hands early

1. In Chapter 2 the input switches from a whole number $n$ to a real number $x$ sliding toward a point. Predict the value of $\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}$ by canceling the common factor, exactly as you’d simplify a sequence ratio. (Next chapter makes this rigorous — and explains why the cancel is legal.)
2. You met $e$ as $\lim\left(1+\tfrac1n\right)^n$. Locate it from the other direction: with a calculator, compute $\tfrac{2^h-1}{h}$ and $\tfrac{3^h-1}{h}$ for small $h$ (say $h=0.001$). One is below $1$, one above. Between which two whole-number bases must the “slope $=1$” base — which Chapter 5 will reveal is $e$ — lie?
3. Your recursive sequences found their limit by demanding $L=(\text{the rule applied to }L)$ — a fixed point. In Chapter 8 you’ll iterate $x \leftarrow x-\tfrac{f(x)}{f'(x)}$ to hunt a root. Predict: if that iteration settles to a value $L$, what equation must $L$ satisfy?

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