MathDB

Problem 3

Part of 2011 VJIMC

Problems(2)

infinite sums equal

Source: VJIMC 2011 1.3

6/1/2021
Prove that ∑k=0∞xk1+x2k+2(1−x2k+2)2=∑k=0∞(−1)kxk(1−xk+1)2\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}for all x∈(−1,1)x\in(-1,1).
Summation
complex rational function

Source: VJIMC 2011 2.3

6/1/2021
Let pp and qq be complex polynomials with deg⁡p>deg⁡q\deg p>\deg q and let f(z)=p(z)q(z)f(z)=\frac{p(z)}{q(z)}. Suppose that all roots of pp lie inside the unit circle ∣z∣=1|z|=1 and that all roots of qq lie outside the unit circle. Prove that max⁡∣z∣=1∣f′(z)∣>deg⁡p−deg⁡q2max⁡∣z∣=1∣f(z)∣.\max_{|z|=1}|f'(z)|>\frac{\deg p-\deg q}2\max_{|z|=1}|f(z)|.
complex analysisalgebrarational functionfunction