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 3 of 4: Finish part (a) with AM-GM

In plain words

The target $3.999$ is just below the limiting constant $4$ .

\frac{(1+T)^2}{T}=T+2+\frac1T\ge4,\quad S_n\ge4\frac{n-1}{n}

Detailed analysis

AM-GM gives $T+1/T\ge2$ . Thus $S_n\ge4(n-1)/n$ . For $n=4000$ , this equals $3.999$ , and it is larger thereafter.