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 2 of 4: Use monotonicity

In plain words

Monotonicity controls the denominator by the middle terms.

T=\sum_{i=1}^{n-1}x_i,\quad x_n\le\frac{T}{n-1},\quad S_n\ge\frac{(1+T)^2}{T+T/(n-1)}=\frac{n-1}{n}\frac{(1+T)^2}{T}

Detailed analysis

Because $x_1\ge\cdots\ge x_n$ , the last term is at most the average of $x_1,\ldots,x_{n-1}$ . Also $x_0=1$ , giving the bound.