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# 38 Questions

 25 Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$ 21 If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$ 7 Consider convergence of series: $\sum_{n=1}^{\infty}\sin\left[\pi\left(2+\sqrt{3}\right)^n\right]$ 7 Determine the value of the integral $I=\int_{0}^{1}\frac{\ln\left(1-a^2x^2\right)}{\sqrt{1-x^2}}dx$ 7 Determine value the following: $L=\sum_{k=1}^{\infty}\frac{1}{k^k}$

# 1,117 Reputation

 +10 calculation of $\int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx$, where $n\in \mathbb{N}$ -2 Computing the Gaussian integral with step functions +20 If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$ +5 Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$

 6 Integral of $\frac{1}{x^4+1}$ 4 Calculation of $\int_{0}^{1}\frac{x-1+\sqrt{x^2+1}}{x+1+\sqrt{x^2+1}}dx$ 4 How to prove $\left(\frac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\frac{n}{n+1}\right)^n$ 4 Proving $\frac{1}{\cos^2\alpha}<\frac{\tan\beta-\tan\alpha}{\beta-\alpha}<\frac{1}{\cos^2\beta}$ 3 Find the determinant, assuming that

# 37 Tags

 27 calculus × 22 4 definite-integrals × 4 12 integration × 19 4 factorial 8 inequality × 2 3 matrices × 4 6 indefinite-integrals × 4 3 linear-algebra × 3 5 limits × 29 3 determinant × 2

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