Problem 3

Consider infinite sequences

\{x_n\}

of positive reals such that

x_0=1

and

x_0\ge x_1\ge x_2\ge\cdots

. (a) Prove that for every such sequence there is an

n\ge1

such that

S_n=\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\ge3.999

. (b) Find such a sequence for which

S_n<4

for all

n

Step 1 of 4: Apply Cauchy-Schwarz

In plain words

Many fractions combine into one ratio of two simple sums.

S_n=\sum_{i=1}^n\frac{x_{i-1}^2}{x_i}\ge\frac{(x_0+\cdots+x_{n-1})^2}{x_1+\cdots+x_n}

Detailed analysis

Cauchy-Schwarz in Engel form gives the displayed lower bound.