Problem 3

Let

1=a_0\le a_1\le a_2\le\cdots

be a sequence of real numbers. Define

b_n=\sum_{k=1}^n\left(1-\frac{a_{k-1}}{a_k}\right)\frac1{\sqrt{a_k}}

. (a) Prove that

0\le b_n<2

for every

n

. (b) Given any

b

with

0\le b<2

, prove that there is such a sequence for which

b_n>b

for infinitely many

n

Step 2 of 5: Interpret rectangles

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In plain words

Rectangles under a decreasing curve give a visual meaning to every summand.

[X_k]=(a_k-a_{k-1})a_k^{-3/2}

Six right-endpoint rectangles under

f(x)=x^{-3/2}

Detailed analysis

Take $X_k$ with base $[a_{k-1},a_k]$ and height $a_k^{-3/2}$ . Since $f(x)=x^{-3/2}$ decreases, $X_k$ lies below the graph and the rectangles have disjoint interiors, so $b_n$ is their total area.