MathDB

3

Part of 1997 IMC

Problems(2)

I think posted before

Source: IMC 1997 day 1 problem 3

10/1/2005
Let A,B∈Rn×nA,B \in \mathbb{R}^{n\times n} with A2+B2=ABA^2+B^2=AB. Prove that if BA−ABBA-AB is invertible then 3∣n3|n.
linear algebralinear algebra unsolved
Sum convergence

Source: IMC 1997 day 2 problem 3

10/19/2005
Show that ∑n=1∞(−1)n−1sin⁡(log⁡n)nα\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha} converges iff α>0\alpha>0.
trigonometrylogarithmsfloor functionfunctionintegrationabsolute valuecomplex numbers