1

Part of 1950 Miklós Schweitzer

Problems(2)

Miklos Schweitzer 1950_1

Source: first round of 1950

Let \{k_n\}_{n \equal{} 1}^{\infty} be a sequence of real numbers having the properties

k_1 > 1

and k_1 \plus{} k_2 \plus{} \cdots \plus{} k_n < 2k_n for n \equal{} 1,2,.... Prove that there exists a number

q > 1

k_n > q^n

for every positive integer

n

algebra proposedalgebra

Miklos Schweitzer 1950_1

Source: second part of 1950

a>0

d>0

and put f(x)\equal{}\frac{1}{a}\plus{}\frac{x}{a(a\plus{}d)}\plus{}\cdots\plus{}\frac{x^n}{a(a\plus{}d)\cdots(a\plus{}nd)}\plus{}\cdots Give a closed form for

f(x)

real analysisreal analysis unsolved