Calculus I · Adding Up Forever

Where this leaves us — and where it goes

Look back at the knot this chapter tied. An infinite series is a limit of partial sums — the sequences of Chapter 1. The benchmark series is the geometric one — the Taylor series of Chapter 7. The deepest tool for convergence is the integral test — the rectangles and areas of Chapters 10 and 11. And the whole apparatus finally pins down where Taylor series live — closing the one question Chapter 7 had to leave open. Nothing here was new machinery; it was every idea in the book, brought together to make sense of the infinite. That is what a capstone should do.

And it opens the last door. We have now added up infinitely many powers of $x$. Book 3 will add up infinitely many waves — the Fourier series glimpsed in Chapters 7 and 10, where a function is built not from $1, x, x^2, \ldots$ but from $\sin x, \cos x, \sin 2x, \ldots$. Deciding when those infinite sums of waves converge will use exactly the convergence ideas you just built — but constructing them, and seeing why waves are the natural building blocks, needs a genuinely new language: vectors, inner products, and orthogonal bases, the machinery of linear algebra. That is where the next book begins, and it is the mathematical ground that modern data analysis, signal processing, and machine learning actually stand on.

You started this book with the slope of a single point. You end it able to add up infinitely many things without fear — to take the infinite, hold it at arm’s length through a limit, and make it give you a finite, exact answer. Every result was built, never received; every loan was repaid. That is calculus, and it is yours now.

Problems

Six tiers. Fluency and Practice are doing* — decide convergence and find radii, using the handful of principles, not a catalog of tests. Concept & counterexample and ◆ Diamond are thinking — the necessary-not-sufficient trap and the proofs. Teach it is teaching. Looking ahead opens Book 3. Throughout, name which principle you’re using (term test, geometric, comparison, integral test, ratio for radius) — that habit matters more than volume here.*

Fluency — decide it, fast

1. Compute the sum of each convergent geometric series, or say it diverges: (a) $\sum_{n=0}^\infty \left(\tfrac13\right)^n$; (b) $\sum_{n=0}^\infty \left(\tfrac{3}{2}\right)^n$; (c) $\sum_{n=1}^\infty \left(\tfrac15\right)^n$; (d) $\sum_{n=0}^\infty \left(-\tfrac12\right)^n$.
2. Apply the term test: which of these diverge because their terms don’t go to $0$? (a) $\sum \tfrac{n}{2n+1}$; (b) $\sum \tfrac1n$; (c) $\sum \tfrac{n^2}{n^2+1}$; (d) $\sum \tfrac{1}{\sqrt n}$. (For the ones the term test can’t rule out, say so — it’s inconclusive, not a pass.)
3. $p$-series: state convergence or divergence and the value of $p$: (a) $\sum \tfrac{1}{n^3}$; (b) $\sum \tfrac{1}{n^{1/2}}$; (c) $\sum \tfrac1n$; (d) $\sum \tfrac{1}{n^{1.01}}$.
4. Comparison: decide convergence by comparing to a geometric or $p$-series: (a) $\sum \tfrac{1}{2^n + n}$; (b) $\sum \tfrac{1}{n^2 + 1}$; (c) $\sum \tfrac{1}{n^2 - n}$ (for $n \ge 2$).
5. Radius of convergence via the ratio test: (a) $\sum \tfrac{x^n}{n!}$; (b) $\sum x^n$; (c) $\sum \tfrac{x^n}{2^n}$; (d) $\sum n!\,x^n$.
6. Write the first four partial sums of $\sum_{n=1}^\infty \tfrac{1}{n(n+1)}$ and guess the limit. (The terms telescope: $\tfrac{1}{n(n+1)} = \tfrac1n - \tfrac1{n+1}$.)

Practice — multi-step

1. Use the integral test to decide $\sum \tfrac{1}{n(\ln n)}$ (for $n \ge 2$). (Set up $\int_2^\infty \tfrac{1}{x\ln x}\,dx$ and substitute $u = \ln x$.)
2. Use the integral test to confirm $\sum \tfrac{1}{n^2}$ converges, by computing $\int_1^\infty \tfrac{1}{x^2}\,dx$. Then use the integral to also confirm $\sum \tfrac1n$ diverges.
3. Find the radius of convergence and state the interval $(-R, R)$ for: (a) $\sum \tfrac{x^n}{n}$; (b) $\sum \tfrac{x^n}{n^2}$; (c) $\sum \tfrac{(x-2)^n}{3^n}$ (centered at $2$ — find where it converges).
4. The geometric series gives $\tfrac{1}{1-x} = \sum x^n$ for $|x|<1$. By substituting, write a series for $\tfrac{1}{1+x^2}$ and state where it converges — then connect it to the arctangent series of Chapter 7.
5. A bouncing ball rises to $\tfrac34$ of its previous height on each bounce, starting from $2$ meters. Find the total vertical distance it travels before coming to rest. (A geometric series — mind the up-and-down.)
6. Decide convergence and name your principle for each: (a) $\sum \tfrac{n}{n^3+1}$; (b) $\sum \tfrac{1}{2^n}$; (c) $\sum \tfrac{1}{\sqrt{n}}$; (d) $\sum \tfrac{n+1}{n}$.

Concept & counterexample — think; true/false with justification, or build the example

1. 1. If $a_n \to 0$, then $\sum a_n$ converges. (b) If $\sum a_n$ converges, then $a_n \to 0$. (c) The harmonic series diverges even though its terms go to $0$. (d) A geometric series $\sum r^n$ converges for every $r$ with $|r| < 1$.
2. The fundamental trap. Explain, using the harmonic series as your example, exactly why “the terms go to zero” is necessary but not sufficient for convergence.
3. Construct a divergent series whose terms go to zero (other than the harmonic series). (A $p$-series with the right $p$ will do.)
4. Find the error. A student writes “$\sum \tfrac1n$ converges because $\tfrac1n \to 0$.” Identify the precise misuse of the term test and give the correct conclusion with a reason.
5. Explain why the integral test requires $f$ to be positive and decreasing — what could go wrong with the rectangle-vs-area comparison if $f$ wandered up and down?
6. Is it possible? A power series with radius of convergence $R = 0$ (it converges only at its center). What would the coefficients have to do? (Look at $\sum n!\,x^n$.)

◆ Diamond — the proofs behind the principles

1. ◆ Derive the geometric series sum $\sum_{n=0}^\infty r^n = \tfrac{1}{1-r}$ for $|r|<1$ by finding a closed form for the partial sum $S_N$ and taking the limit. State exactly where $|r|<1$ is used.
2. ◆ Prove the term test: if $\sum a_n$ converges, then $a_n \to 0$. (Write $a_n = S_n - S_{n-1}$.) Then explain why the converse is false in one sentence.
3. ◆ Prove that the harmonic series diverges using Oresme’s grouping argument (blocks of doubling length, each exceeding $\tfrac12$). Write out the first three blocks explicitly.
4. ◆◆ Prove the integral test by establishing the rectangle inequalities $\sum_{n=2}^\infty f(n) \le \int_1^\infty f \le \sum_{n=1}^\infty f(n)$ for positive decreasing $f$, and conclude the sum and integral converge together. (Draw the two rectangle pictures — under and over the curve.)
5. ◆◆ Prove the $p$-series result ($\sum \tfrac{1}{n^p}$ converges iff $p > 1$) by applying the integral test and the value of $\int_1^\infty x^{-p}\,dx$ from Chapter 11. Handle $p = 1$ separately.
6. ◆◆ Use the ratio test to prove $e^x = \sum \tfrac{x^n}{n!}$ converges for every real $x$ ($R = \infty$), and reconcile this with the remainder argument you gave in Chapter 7 — two different proofs of the same fact.

Teach it — explain it, write it up, or fix it

1. In three or four sentences, teach a friend what it means for an infinite series to converge — that it’s the limit of partial sums, nothing more exotic — using $\tfrac12 + \tfrac14 + \tfrac18 + \cdots = 1$.
2. Debug your classmate. They claim $1 + \tfrac12 + \tfrac13 + \tfrac14 + \cdots$ converges “because each term is smaller than the last and they’re heading to zero.” Give them the one fact that demolishes this, in a sentence.
3. Write a complete worked solution deciding whether $\sum \tfrac{1}{n^2 + 1}$ converges, naming your comparison series and justifying the inequality, so a peer could do $\sum \tfrac{1}{n^3 + 5}$.
4. In a few sentences, teach the integral test as a picture — how rectangles drawn against a curve trap a sum between two copies of an area — and why that means a sum and an integral share the same fate.
5. Design a problem. Invent a series whose convergence is best decided by the integral test, and write the full solution key, including the substitution in the integral.

Looking ahead — Book 3, and the waves to come

1. You can sum powers of $x$ into functions (Taylor) and decide when those sums converge (this chapter). Book 3 sums waves — $\sin x, \cos x, \sin 2x, \ldots$ — into functions (Fourier). Predict one reason waves might be better building blocks than powers for a repeating signal like a sound or a heartbeat. (Hint: what do $\sin$ and $\cos$ do that $x^n$ never does?)
2. The remarkable fact $\sum \tfrac{1}{n^2} = \tfrac{\pi^2}{6}$ is proved with Fourier series. Knowing only that the sum converges (this chapter) and that the answer involves $\pi$, speculate why a sum of pure number-fractions might secretly know about circles. (No proof expected — just notice that $\pi$ appearing here is strange and wonderful, and Book 3 explains it.)

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