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\nEvaluate $\\lim_{n\\to\\infty}n!+n^{2}-\\frac{n}{6}$
\nEvaluate\n$$\\lim_{n\\to\\infty}n!+n^{2}-\\frac{n}{6}$$\nSorry for my bad english
\nA:
\n$$\\sum_{k=0}^\\infty \\frac{(-n)^k}{6^k}= \\frac{n}{6}(1+\\frac{1}{n}+\\frac{1}{n^2}+\\frac{1}{n^3}+\\ldots)$$\n$$= \\frac{n}{6}(1+\\frac{1}{n}(1+\\frac{1}{n}+\\frac{1}{n^2}+\\ldots))=\\frac{n}{6}(1+\\frac{1}{1-\\frac{1}{n}})$$\nNow for $x>1$\n$$\\frac{1}{1-x}=1+x+x^2+\\ldots$$\n$$\\frac{1}{1-x}>1+x+x^2+\\ldots$$\n$$\\frac{1}{1-x}-1>x+x^2+x^3+\\ldots$$\n$$\\frac{x}{1-x}>x+x^2+x^3+\\ldots$$\n$$\\frac{x}{1-x}>x\\left(1+\\frac{1}{x}+\\frac{1}{x^2}+\\ldots\\right)=x+x^2+x^3+\\ldots$$\nwhich can only hold when $x\\lt1$.\nSo\n$$\\sum_{k=0}^\\infty \\frac{(-n)^k}{6^k}= \\frac{n}{6}(1+1+\\frac{1}{n}+\\frac{1}{n^2}+\\ldots)$$\nor\n7582aa13b2\n.0.2.13 ØµØ±Ø ØªØ¯ÙŠ Ø®Ùد واØير بروالق سيرÙر على٠و علىÙ.\nIt provides a reliable solution for editing any supported RAW file. Here you can get their keygen at the cheapest price. you can download their keygen from various websites.
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