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 4 of 5: Choose an arithmetic sequence

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In plain words

Equal spacing turns the rectangles into the textbook Riemann sum.

a_k=1+k\\Delta x\\quad(\\Delta x>0)

Detailed analysis

With $a_k=1+k\Delta x$ , $b_n$ is the right-endpoint Riemann sum for $\int_1^{a_n}x^{-3/2}dx$ ; holding the endpoint fixed and taking $\Delta x\to0^+$ makes the sum converge to the integral.