Jace holds his thumb over the air for one beat—quality gets a vote.
"Accepted," he says, and looks Jordan in the eye so the contract feels real. "**Listen-only. One Q&A window later. Transfer upfront. If this derails, I refund $30 and book you separate."
"Yes," Jordan says, already thumbing.
Jace nods once.
Her screen lights green; his phone kicks.
[SYSTEM PROMPT] Income detected: $60.00.[SYSTEM PROMPT] Evaluating Talent…[SYSTEM PROMPT] Money Welfare: ×3.[SYSTEM PROMPT] Disbursement today: +$180.00.[SYSTEM PROMPT] Total money crit disbursed today: +$6,910.00.[SYSTEM PROMPT] Daily cap remaining (Money): $93,090.00.
He verifies the name—Jordan L.—gives the human nod, and flips the page.
"Clock check," he says. He writes T–60 in the corner. "We do: Alternating Series Test (AST), Ratio/Root, Integral Test, quick tour of p-series and comparison traps. One demo, one you, then Q&A window."
Eva's pencil stands at attention. Jordan sits down with a vow in her posture.
"AST," Jace says, and draws a neat alternating sum: ∑ (−1)^{n+1} (1/(n+1)). "Three rails: alternating, decreasing in magnitude eventually, limit → 0. If all hold, the series converges. Error after N terms bounded by the first neglected term's magnitude."
He underlines decreasing twice. "This is where profs eat you. Show the monotone decline, don't mumble it."
He writes a_{n} = 1/(n+1). "Monotone?" he asks.
Eva: "Yes—a_{n+1} = 1/(n+2) < a_n. And lim a_n = 0."
"Converges by AST," Jace says. He writes |error| ≤ a_{N+1} in a little box and taps it. "This box saves points."
"Demo me a non-example," Jordan whispers, then clamps her lips because she promised silence. Max lifts two fingers: later.
"Ratio test," Jace says, and prints ∑ (3^{n}/n!). "You'll see this. Compute L = lim |a_{n+1}/a_n|."
Eva writes: a_{n} = 3^{n}/n!, so a_{n+1}/a_n = (3^{n+1}/(n+1)!)/(3^{n}/n!) = 3/(n+1) → 0. "L = 0 < 1 so converges absolutely."
"Root test?" Jace says, and writes ∑ ( (2n)/(3n+1) )^{n}.
She breathes through it and writes lim √[n]{|a_n|} = lim (2n/(3n+1)) = 2/3 < 1. "Converges absolutely."
"Integral test," he says, drawing ∑ 1/(n(ln n)^2) starting at n = 2. "Classic trap. f(x) = 1/(x(ln x)^2) decreasing and positive for x ≥ 2; integrate: ∫ 2→∞ 1/(x(ln x)^2) dx. u-sub u = ln x, du = dx/x, gives ∫ (1/u^2) du which is −1/u |→∞ = 1/ln 2. Finite → converges."
Eva's eyes widen. "I always mis-remember that one. I think it diverges because the denominator looks small."
"That's why we write it out," Jace says. He boxes u = ln x and draws a tiny arrow: du = dx/x. "The trap there is gut. We cage gut with work."
Jordan has been writing like a court reporter with better hair.
"Your turn," Jace says to Eva. He prints ∑ ( (−1)^{n} / √n ) and leans back.
She checks rails. "Alternating? Yes. Magnitudes decreasing? 1/√n decreases for n ≥ 1. Limit to 0? Yes." She writes converges by AST and then thinks aloud. "Absolutely?∑ 1/√n diverges (p-series with p = 1/2). So conditionally convergent."
"Flag that word," Jace says. He writes conditionally in block letters and underlines it twice. "Graders see that as a sign you've met the world before."
He pivots. "Comparison—show me something easy. ∑ 2/(n^2 + 4)."
"Compare to 2/n^2," she says. "Since n^2 + 4 ≥ n^2, 2/(n^2 + 4) ≤ 2/n^2, and ∑ 2/n^2 converges, so ours converges by Direct Comparison."
"Good," he says. "Now the Limit Comparison trap." He writes ∑ ( (n + 5)/(n^2 + 1) ).
"Compare with 1/n," she says, pushing it. "lim ( (n + 5)/(n^2 + 1) ) / (1/n) = lim ( (n + 5)/ (n^2 + 1) * n ) = lim ( (n^2 + 5n)/(n^2 + 1) ) = 1, so both behave the same; ∑ 1/n diverges → diverges."
"Clean," Jace says. "Keep the algebra honest. If it goes to a positive finite number, the twins share a fate."
Max taps the table twice: T–30.
Jace draws a small clock. "Q&A block: five minutes," he says. He nods to Jordan: "One question."
She exhale-smiles like she just got a hall pass. "Alternating harmonic partial sums: is there a fast way to estimate how many terms to get error < 0.01?"
"AST error bound," Jace says, tapping the earlier box. "We need a_{N+1} = 1/(N+1) < 0.01 → N+1 > 100 → N ≥ 100. One hundred terms gets error under a cent."
Jordan writes 100 with the triumph of a person who got to keep a number.
"Window closed," Jace says, pleasant but locked. "Back to Eva."
They run a ratio-test divergence example (a cheeky L = 1 that tells you nothing) and choose a different test; they swat one Root test where L > 1 and the verdict is diverge; they revisit p-series and he draws a tiny ladder: p > 1 → converge; p ≤ 1 → diverge. He prints Don't mix tests in the corner, because the graders appreciate scolding in the right direction.
Max taps T–10 and slides a water bottle into the frame like a stagehand.
"Last ten," Jace says. He flips to a fresh half-page and titles it TRAP SHEET in block caps. Below:
Signs: box the minus in IBP; say it out loud.
Constants:jail the 1/2; track it line to line.
Pick tests by anatomy:alternating → AST, factorials/powers → Ratio, nth power blobs → Root, 1/(x (ln x)^k) → Integral (u = ln x), polynomials → p-series/comparison.
Absolute vs Conditional: show both if alternating.
Error Bound (AST):|error| ≤ a_{N+1}.
Limit Comparison: compute lim a_n/b_n clean; if 0 < L < ∞, twins share fate.
Don't double-test. If one test settles it, stop.
He tears the page with the neat rip of a man who has practiced and hands it to Eva. "Two-minute ritual," he says. "Water. Bathroom. Three deep breaths. Write 'signs, constants, pick test' at the top of your scratch before you touch the first series. Then let momentum be dumb for you."
Eva takes the page like it's laminated. "I… can breathe," she says, stunned. "I can actually breathe."
"You'll do fine," Jace says. It isn't flattery. It is an observation.
"Tutor rate for a future session?" Jordan asks, still within rules.
"Same rails," Jace says. "Tomorrow noon, 60 minutes, $80. Transfer upfront. Public space."
"Yes," Jordan says, already opening her calendar like a religious text. "I'll DM."
Max checks the sky with the authority of a weatherman. "Quiz at four?" he says.
Eva nods. "Room 214. They love ambush questions."
"You love trap sheets," Jace says. "We're even."
They stand. The drumline slides off toward the stadium, leaving air that has been previously drummed. The shade surrenders them to a day that feels less expensive.
Eva points at the constants jail with a grin. "This is stupid," she says fondly. "I love it."
"Stupid and repeatable is design," Jace says.
She salutes with the paper, mouths thanks to Max, and goes—feet faster now, shoulders lower, pencil doing the optimistic click of a creature that thinks it will be used well.
Jordan lingers one beat longer, then taps send on a calendar invite to tomorrow noon, and power-walks after her roommate like a second thought that decided to be first.
Quiet lands. It isn't empty. It is done.
Jace writes one line in his wallet notebook because paper makes pride into inventory. "Tutoring: $120×2 → +$240; add-on $60×3 → +$180; Money +$6,910; cap $93,090." He closes the book and lets the math stay math.
Max flips the bottle cap closed and watches the path like it might confess something. "Are we heroes?"
"We are procedures," Jace says, which is the flavor of heroism he can digest.
His phone hums. Not Sandra. Not Taj. Mara—campus Event Coordinator he's seen on flyers and tech sheets—mutual with the radio booth and Sara-the-green-hoodie.
Mara (Events): hey, heard you run a tidy money brain. need a 60-min Budget Bootcamp tonight for orientation stragglers. $200, transfer after, classroom C109. 7 pm? we just need rails + Q&A. interested?
Max's eyebrows climb like a cat on curtains. "The universe tips," he says.
Jace stares at the screen and lets time join the conversation. "Rails check," he says, half to himself, half to Max. "New merchant: campus payments.Transfer after delivery—fine. Pacing—today already had store plus tutoring. Attention—low; classroom. Fatigue—real; but content is procedure I can recite in my sleep."
"Do you want to recite it in your sleep," Max says, "or do you want to sleep?"
Jace flips the phone so it lies flat and looks at the envelope instead. NO RE-ENTRY sits on the corner like a guard dog. This isn't re-entry. It's a microphone. He could do slides on a napkin. He could also not.
The panel floats a calm note at the top of his vision, not pushy, just present.
[SYSTEM PROMPT] Advisory: session length moderate. Overnight recovery improves variance.
"I hear you," Jace says. He lifts the phone again. The message box waits like a courteous doorway. He drafts two responses:
(A)Tonight 7 pm works. Transfer after is fine. I'll bring rails + Q&A.(B)Tomorrow after 10 is better. Same terms. We'll fill the room and give you better heads.
He holds between them with his thumb hovering, and lets want and plan try to win without lying.
Max watches his face like a sunrise he didn't pay for. "We can also eat," he says. "Humans do that."
"Humans eat," Jace echoes, amused.
He doesn't press. Not yet.
He breathes, and the day breathes with him.