fix: proofread Calculus 101 part

[vEnhance]

This proofreads the Calculus 101 part (the first unchecked box in #315), covering the five chapters under tex/calculus/: limits.tex, p-adic.tex, differentiate.tex, taylor.tex, integrate.tex.

All findings were minor (typos, grammar, and small index slips). No large mathematical errors were found.

limits.tex

• Comma splice "…already been done, / For example" → period.
• "if $S$ has not bounded above/below" → "is not bounded".
• Malformed prototype sum \sum_{k \ge 1}^\infty → \sum_{k=1}^\infty.
• "you can use essentially copy its proof" → drop stray "use".
• "where in general we stuck on the decimal representation" → "tack".
• "the last fourth condition" → "the last condition".
• Index typo in the alternating series test solution: (a_{N-2} - a_{N+1}) → (a_{N-2} - a_{N-1}).

p-adic.tex

• "close on the number of line" → "number line".
• "$\QQ_p$ and $\ZZ_p$ becomes a metric space" → "become metric spaces".
• "Last post we proved" → "Earlier we proved" (leftover blog phrasing).
• Skolem–Mahler–Lech proof: "Fix any $0 \le r &lt; N$" used an undefined $N$ → "$< T$".
• Mahler-form expansion of $(\id+pB)^n$: summation terms were indexed by $n$ instead of the summation variable $k$ (p^n, B^n, a_n → p^k, B^k, a_k).
• "Strassman's theorem" → "Strassmann's" (matches the earlier spelling in the same chapter).

differentiate.tex

• "as a function it its own right" → "in its own right".
• Duplicated "then then" in the sum and product rule statements.
• Singular "local extrema" → "local extremum" (3 spots).
• "manually check all cases outside $Z$" → "outside $S$" (the interior; boundary points lie in $Z\setminus S$).
• "The one is simple" → "The first one is simple".
• Stray word: "it lets you relate use derivative information…" → drop "relate".
• Extra comma in "secant joining $(a,f(a))$, to $(b,f(b))$".
• "which we'll grow more next time" → "which we'll see more of next time".

taylor.tex

• "It terms out" → "It turns out".
• Stray period inside the radius-of-convergence display before "with the convention".
• "$p(x) = x^3+7x+9$" → "$P(x)$" to match the surrounding text.

integrate.tex

• "This difference is that…" → "The difference is that…".
• "a typical a rectangle function" → drop extra "a".
• "What we have done is taking … and showed" → "is take … and show".
• "a point $\xi_i$ in the $n$th interval" → "$i$th interval".
• "As in the proof \Cref{…}" → "As in the proof of \Cref{…}".
• "the meshes … approaches zero" → "approach zero".

🤖 Generated with Claude Code

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Generated by Claude Code