Let p(x)=2x^6+4x^5+3x^4+5x^3+3x^2+4x+2. Let \displaystyle I_{k}=\int^{\infty}_{0}\frac{x^k}{p(x)}dx
where 0<k<5. Then value of k for which \displaystyle I_{k} is smallest.
Thanks friends for yours fantastic solutions
i have solved it using Integration.
after seeing above solutions by opalg and klass van have seems that my solution is partial (Not fully satisfactory)
Solution Put
\begin{equation*}
I_{n}=\sqrt{n}\int_{0}^{\pi/4}\cos^{2n-2}(x)\,\mathrm{d}x.
\end{equation*}
Via the substitutions $ y=\sin x $ and $ y=\frac{z}{\sqrt{n-1}} $ we get
\begin{gather*}
I_{n}=\sqrt{n}\int_{0}^{\pi/4}(1-\sin^2(x))^{n-1}\,\mathrm{d}x =...
Solution Let $\displaystyle I = \int\sqrt{\frac{(1-\sin x)(2-\sin x)}{(1+\sin x)(2+\sin x)}}dx$
We can write $\displaystyle \sqrt{\frac{1-\sin x}{1+\sin x}} = \sqrt{\frac{1-\sin x}{1+\sin x}\times \frac{1+\sin x}{1+\sin x}} = \frac{\cos x}{1+\sin x}$So we get $\displaystyle I = \int\frac{\cos...
Thanks Ifdahl for nice solution
Here is mine
Using $n!=n(n-1)! = [(n-1)+1](n-1)! = (n-1)(n-1)!+(n-1)(n-2)!$
So $(n-1)(n-1)!+(n-1)(n-2)!<(n-1)(n-1)!+(n-1)(n-2)!+(n-1)(n-3)!+\cdots (n-1)1!\;\forall n\geq 4$
So $n!<(n-1)\bigg[(n-1)!+(n-2)!+(n-3)!+\cdots +2!+1!\bigg]\cdots \cdots (1)$
And...
Consider the sequence $a_{n}$ given by $\displaystyle a_{1} = \frac{1}{3}$ and $\displaystyle a_{k+1}=a^2_{k}+a_{k}$ for $k\geq 2$and Let $\displaystyle S = \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots \cdots +\frac{1}{a_{2008}}$. Then $\lfloor S \rfloor $ is