Some Interesting Problems in Calculus

\text{1)Evaluate:}

a)\lim_{n\rightarrow \infty}\frac{1}{2n}\log\binom{2n}{n}

b)\lim_{n\rightarrow \infty}\left[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}\right]

\text{2)Let } a_{1}=1 \text{ and }a_{n}=n(a_{n-1}+1)\ \forall n\geq 2 .\text{Find }

\lim_{n\rightarrow \infty}\left(1+\frac{1}{a_{1}}\right)\left(1+\frac{1}{a_{2}}\right)\cdots \left(1+\frac{1}{a_{n}}\right)

\text{3)Let }a_0 =0<a_1<a_2<\cdots <a_n \text{ be real numbers.}
\text{Suppose }p(t)\text{ is a real valued polynomial of degree n such that}

\int\limits^{a_{j+1}}_{a_{j}}p(t)\, dt=0 \qquad \text{ for all } 0\leq j \leq n-1

\text{Show that for }0\leq j\leq n-1 \text{ the polynomian p(t)has }
\text{exactly one root in the interval }(a_j,a_{j+1})

\text{4)Let f(u) be a continuos function and,let \[u\]}\rightarrow GIF
\text{Show that for any }x>1
\int \limits ^{x}_{1}\[u\](\[u\]+1)f(u)\, du=2\sum\limits ^{\[x\]}_{i=1}i\int \limits^{x}_{i}f(u)\, du

\text{5)Let f(x) be a function differentiable n+1 times for some }n>0

f(1)=f(0)=f^{(0)}(0)=\cdots =f^{(n)}(0)=0

\text{Prove that }f^{(n+1)}(x)=0 \text{ for some }x\in (0,1)

\text{6)Prove that for }n>0:

\int \limits ^{\pi/2}_{0}\frac{\sin(2n+1)x}{\sin x}\, dx=\frac{\pi}{2}

\text{7)Show that}

\int \limits^{\pi}_{0}\left|\frac{\sin nx}{x}\right|\, dx \geq \frac{2}{\pi}\left(1+\frac{1}{2}+\cdots +\frac{1}{n}\right)

\text{8)Using the identity }\log x = \int^x_1\frac{dt}{t}\qquad x>0 \text{ ,show that for }n\geq 1

\frac{1}{n+1}\leq \log\left(1+\frac{1}{n}\right)\leq \frac{1}{n}

\text{9)Let }a_1>a_2>a_3\cdots >a_r>0

\text{Compute }\lim_{n \rightarrow \infty}(a_1^n+a_2^n+\cdots +a_r^n)^{1/n}

\text{10)Find the maximum value of }\cos\alpha_1\cdot\cos\alpha_2\cdots \cos \alpha_n
\text{under the restriction }0\leq \alpha_1,\alpha_2,\cdots ,\alpha_n\leq \frac{\pi}{2} \text{ and }}
\cot\alpha_1\cdot\cot\alpha_2\cdots \cot \alpha_n =1.