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26.設\(\displaystyle \lim_{n \to \infty}\sum_{k=1}^n \frac{1}{\sqrt{k^2+nk+n^2}}=\ln\alpha\)，則\(\alpha\)之值為何？
(A)\(\displaystyle 1+\frac{\sqrt{3}}{3}\) (B)\(\displaystyle 1+\frac{2\sqrt{3}}{3}\) (C)\(\displaystyle 1+\sqrt{3}\) (D)\(\displaystyle 1+\sqrt{2}\)
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[解答]
\(\displaystyle \lim_{n\to \infty}\sum_{k=1}^n \frac{1}{n}\frac{1}{\sqrt{\left(\frac{k}{n}\right)^2+\left(\frac{k}{n}\right)+1}}=\int_0^1 \frac{1}{\sqrt{x^2+x+1}}dx=asinh\left(\frac{2x+1}{\sqrt{3}}\right)\Bigg\vert\;_0^1=asinh(\sqrt{3})-asinh(\frac{1}{\sqrt{3}})\)
\(\displaystyle =\ln(\sqrt{3}+2)-\ln(\frac{1}{\sqrt{3}}+\frac{2}{\sqrt{3}})=\ln\left(\frac{\sqrt{3}+2}{\frac{3}{\sqrt{3}}}\right)=\ln\left(1+\frac{2}{\sqrt{3}}\right)=\ln \alpha\)，\(\alpha=1+\frac{2}{\sqrt{3}}\)
\(\displaystyle \int \frac{1}{x^2+x+1}dx=asinh\left(\frac{2x+1}{\sqrt{3}}\right)+c\)積分過程只好借助網站幫忙計算
[url]https://www.emathhelp.net/en/calculators/calculus-2/integral-calculator/?f=1%2Fsqrt%28x%5E2%2Bx%2B1%29&var=x[/url]
公式\(arsinhx=ln(x+\sqrt{x^2+1})\)，\(-\infty<x<\infty\)
[url]https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions[/url]
類似題
求極限\(\displaystyle \lim_{n\to \infty}\sum_{k=0}^{n-1}\frac{1}{\sqrt{n^2+k^2}}\)。
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30.
設\(x=\root 3\of{\sqrt{17}+3}-\root 3\of{\sqrt{17}-3}\)，則\(x^3+6x+7\)之值為何？
(A)10 (B)11 (C)12 (D)13
37.
設\(A=\left(\matrix{4&4\cr -1&-1}\right)\)，而\(A+A^2+\ldots+A^n=\left(\matrix{2(3^n-1)&a \cr b&c}\right)\)，求\(b+c=\)？
(A)\(3^n-1\) (B)\(1-3^n\) (C)\(4(3^n-1)\) (D)\(4(1-3^n)\)
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