MathDB

1

Part of 1997 IMC

Problems(2)

limit of a sequence

Source: IMC 1997 day 1 problem 1

10/1/2005
Let {ϵn}n=1∞\{\epsilon_n\}^\infty_{n=1} be a sequence of positive reals with lim⁡n→+∞ϵn=0\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0. Find lim⁡n→∞1n∑k=1nln⁡(kn+ϵn) \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right)
limitlogarithmscalculusintegrationreal analysisreal analysis unsolved
Show that g is bounded

Source: IMC 1997 day 2 problem 1

10/19/2005
Let f∈C3(R)f\in C^3(\mathbb{R}) nonnegative function with f(0)=f′(0)=0,f′′(0)>0f(0)=f'(0)=0, f''(0)>0. Define g(x)g(x) as follows: {g(x)=(f(x)f′(x))′forx≠0g(x)=0forx=0 \{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 \\ g(x)=0 &\text{for}& x=0\end{array} (a) Show that gg is bounded in some neighbourhood of 00. (b) Is the above true for f∈C2(R)f\in C^2(\mathbb{R})?
functioncalculusderivativereal analysisreal analysis unsolved