[{"_1":2,"_3":-5,"_4":-5},"loaderData",{"_5":6,"_7":8},"actionData","errors","root",{"_26":27,"_28":29,"_30":12,"_31":12,"_32":12,"_33":29,"_34":12,"_35":12,"_36":12,"_37":12,"_38":-5,"_39":40},"routes/canvas.shared.$sharedTextdocId",{"_9":10,"_11":12},"sharedTextdoc",{"_13":14,"_15":-7,"_16":17,"_18":19,"_20":21,"_22":23,"_24":12,"_25":12},"isAuthenticated",false,"sharedTextdocId","697e4e79e3b4819186be19d2c018f647","versionInt","title","Convergence Of (1 + 1/n)^n To E — Interactive Visualization","type","code/react","content","import React, { useMemo, useState } from \"react\";\nimport { LineChart, Line, XAxis, YAxis, CartesianGrid, Tooltip, ReferenceLine, ResponsiveContainer } from \"recharts\";\n\n// --- Math helpers -----------------------------------------------------------\n// Fallback for Math.log1p on older runtimes\nconst log1p = (x) => (Math.log1p ? Math.log1p(x) : Math.log(1 + x));\n\n// Compute (1 + 1/n)^n in a numerically stable way\nfunction pow1pOverN(n) {\n if (!Number.isFinite(n) || n <= 0) return NaN;\n return Math.exp(n * log1p(1 / n));\n}\n\n// Binomial coefficients row using BigInt for exact integer coefficients\nfunction binomialRow(n) {\n const row = [];\n let c = 1n; // C(n,0)\n row.push(c);\n for (let k = 1n; k <= BigInt(n); k++) {\n c = (c * (BigInt(n) - k + 1n)) / k; // C(n,k) from C(n,k-1)\n row.push(c);\n }\n return row; // length n+1\n}\n\nfunction formatBigInt(x) {\n return x.toString();\n}\n\nfunction Fraction({ num, den }) {\n return (\n \n {num}\n {den}\n \n );\n}\n\n// Numeric reconstruction from symbolic binomial (for tests / cross-checks)\nfunction valueFromBinomial(n) {\n const coeffs = binomialRow(n);\n let s = 0;\n for (let k = 0; k <= n; k++) {\n // Safe for small n; tests use modest n only\n s += Number(coeffs[k]) / Math.pow(n, k);\n }\n return s;\n}\n\n// --- Lightweight self-tests -------------------------------------------------\nfunction runSelfTests() {\n const e = Math.E;\n const tests = [];\n\n // 1) Exact small-n sanity\n tests.push({\n name: \"n=1 equals 2\",\n pass: Math.abs(pow1pOverN(1) - 2) < 1e-15,\n detail: `value=${pow1pOverN(1)}`,\n });\n\n // 2) Monotone increasing for sample ns\n const ns = [1, 2, 3, 5, 10, 20, 50];\n let mono = true;\n for (let i = 0; i < ns.length - 1; i++) {\n const a = pow1pOverN(ns[i]);\n const b = pow1pOverN(ns[i + 1]);\n if (!(b >= a)) {\n mono = false;\n break;\n }\n }\n tests.push({ name: \"Monotone increasing (sample)\", pass: mono, detail: `ns=${ns.join(',')}` });\n\n // 3) Always below e for finite n (sample)\n const belowE = ns.every((k) => pow1pOverN(k) < e);\n tests.push({ name: \"Below e for finite n (sample)\", pass: belowE, detail: `e≈${e}` });\n\n // 4) Approaches e (n=1000 within ~1e-4)\n const v1000 = pow1pOverN(1000);\n tests.push({ name: \"n=1000 close to e\", pass: Math.abs(v1000 - e) < 1e-4, detail: `|v-e|=${Math.abs(v1000 - e)}` });\n\n // 5) Binomial row structure (n=5): length n+1 and ends are 1\n const row5 = binomialRow(5);\n tests.push({ name: \"binomialRow(5) length 6\", pass: row5.length === 6, detail: `len=${row5.length}` });\n tests.push({ name: \"binomialRow(5) ends are 1\", pass: row5[0] === 1n && row5[5] === 1n, detail: `${row5[0]}..${row5[5]}` });\n\n // 6) Reconstruct from binomial equals pow1pOverN for small n\n const chkNs = [2, 5, 10];\n const reconOk = chkNs.every((k) => Math.abs(valueFromBinomial(k) - pow1pOverN(k)) < 1e-12);\n tests.push({ name: \"Reconstruction matches numeric (n=2,5,10)\", pass: reconOk, detail: `ns=${chkNs.join(',')}` });\n\n return tests;\n}\n\nfunction Badge({ ok }) {\n return (\n {ok ? \"PASS\" : \"FAIL\"}\n );\n}\n\n// --- App --------------------------------------------------------------------\nfunction App() {\n const [maxN, setMaxN] = useState(200);\n const [n, setN] = useState(10);\n const e = Math.E;\n\n const data = useMemo(() => {\n const arr = [];\n for (let k = 1; k <= maxN; k++) {\n const val = pow1pOverN(k);\n arr.push({ n: k, value: val, error: Math.abs(val - e) });\n }\n return arr;\n }, [maxN]);\n\n const current = useMemo(() => {\n const value = pow1pOverN(n);\n const error = value - e;\n const fracNum = n + 1;\n const fracDen = n;\n\n // Symbolic binomial: ((n+1)/n)^n = sum_{k=0}^n C(n,k) / n^k\n const coeffs = binomialRow(n);\n const terms = coeffs.map((Ck, k) => ({ Ck, k, denExp: k }));\n\n return { value, error, fracNum, fracDen, coeffs, terms };\n }, [n]);\n\n const tests = useMemo(() => runSelfTests(), []);\n\n return (\n

Convergence of (1 + 1/n)ⁿ to e

Explore how (1 + 1/n)ⁿ approaches Euler's number as n increases. Move the sliders and inspect exact fractional forms and the binomial expansion.

\n Max N for chart: {maxN}\n setMaxN(parseInt(e.target.value))} className=\"w-full\"/>\n

The chart plots values for n = 1 … {maxN}.

\n Focus N: {n}\n setN(parseInt(e.target.value))} className=\"w-full\"/>\n

Inspect exact forms for this n.

\n\n {/* Self-test section */}\n

Built-in Checks

Quick sanity tests to ensure the math and rendering behave as expected.

\n \n {t.name}\n {t.detail}\n

\n\n

Exact forms for n = {n}

Base fraction:\n ( 1 + )ⁿ = ( )ⁿ\n

Decimal value: {Number.isFinite(current.value) ? current.value.toPrecision(12) : \"—\"}

Error vs e: {Number.isFinite(current.value) ? (current.value - e).toExponential(6) : \"—\"} {Number.isFinite(current.value) ? (current.value > e ? \"(above e)\" : \"(below e)\") : null}

\n\n

Binomial expansion

Using ((n+1)/n)ⁿ = Σ_k=0^{n} C(n,k)·(1/n)^k = Σ C(n,k) / n^k

\n {current.terms.map((t, idx) => (\n \n {formatBigInt(t.Ck)}{t.k > 0 ? <>/n^{t.denExp} : null}\n {idx < current.terms.length - 1 ? + : null}\n \n ))}\n

Shown exactly with integer coefficients C(n,k) and integer powers n^k.

\n\n

Convergence chart (value vs n)

\n \n \n \n \n \n [typeof v === 'number' ? v.toPrecision(8) : v, name]} />\n \n \n \n \n

Dashed line marks e ≈ {e.toPrecision(12)}. The sequence is increasing and converges to e.

\n\n

Absolute error |(1 + 1/n)ⁿ − e|

\n \n \n \n \n \n [typeof v === 'number' ? Number(v).toExponential(3) : v, name]} />\n \n \n \n

Log-scale shows how the error shrinks with larger n.

\n\n \n

\n );\n}\n\nexport default App;\n","access","public","isModerationBlocked","isAnonifyApiKeyDetected","dd",{"_41":42,"_43":44},"disablePrefetch",true,"shouldPrefetchAccount","shouldPrefetchUser","shouldPrefetchSystemHints","manageRootRouteStylesheet","mobileWebHandoffEnabled","promoteCss","disableSSR","stripModulepreloadImports","paidCampaignAppRedirect","statsigGateEvaluationsPromise",["P",40],"traceId","2360345051323855821","traceTime",1789573940575]

Built-in Checks

Exact forms for n = {n}

Binomial expansion

Convergence chart (value vs n)

Absolute error |(1 + 1/n)n − e|

Absolute error |(1 + 1/n)ⁿ − e|