MathDB

4

Part of 2016 VJIMC

Problems(2)

Nested infinite sum

Source: VJIMC, 26th annual, Category I, Problem 4

4/10/2016
Find the value of sum ∑n=1∞An\sum_{n=1}^\infty A_n, where An=∑k1=1∞⋯∑kn=1∞1k121k12+k22⋯1k12+⋯+kn2.A_n=\sum_{k_1=1}^\infty\cdots\sum_{k_n=1}^\infty \frac{1}{k_1^2}\frac{1}{k_1^2+k_2^2}\cdots\frac{1}{k_1^2+\cdots+k_n^2}.
real analysisVojtech Jarnikinfinite sum
Bounded variation

Source: 26th annual VJIMC (2016), Category II, Problem 4

4/10/2016
Let f:[0,∞)→Rf: [0,\infty) \to \mathbb{R} be a continuously differentiable function satisfying
f(x)=∫x−1xf(t)dtf(x) = \int_{x - 1}^xf(t)\mathrm{d}t
for all x≥1x \geq 1. Show that ff has bounded variation on [1,∞)[1,\infty), i.e.
∫1∞∣f′(x)∣dx<∞.\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.
functionIntegralbounded variationdifferentiable functionreal analysiscollege contests