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 4 of 4: Construct the sequence for part (b)

In plain words

The ratio $1/2$ makes the successive summands $2,1,1/2,\ldots$ , approaching but never reaching $4$ .

x_i=2^{-i}\quad\Longrightarrow\quad S_n=2+1+\frac12+\cdots+2^{2-n}=4-2^{2-n}<4

Detailed analysis

The sequence is positive, nonincreasing, and starts at $x_0=1$ . Direct substitution gives the geometric sum shown, which is always less than $4$ .