×

# An interesting inequality

Note by Vivek V P
4 years, 11 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

So, to solve this problem, a substitution must be made. Let $y_i = \frac{1}{2} - x_i$. Therefore $0 < y_i < \frac{1}{2}$ for all $i$. Also of note, $\frac{1}{2} + x_i = 1 - y_i$ is easily verified. By this definition, $a_1 = 1 - y_1, a_2 = (1 - y_2) y_1, a_3=(1 - y_3) y_2 y_1,...$ or in general, $a_i = (1 - y_i) \displaystyle \prod_{i=1}^{i - 1} y_i = \displaystyle \prod_{i=1}^{i - 1} y_i - \displaystyle \prod_{i=1}^i y_i$. I'll let the reader explore why the forth statement given isn't necessary. So from this, $\displaystyle \sum_{k=1}^\infty ka_k = 1 - y_1 + 2y_1 - 2y_1y_2 +3y_1y_2 - 3y_1y_2y_3 + 4y_1y_2y_3y_4 \ldots = 1 + y_1 + y_1y_2 + y_1y_2y_3 + \ldots$. The only way to maximize the sum is to have every $y_i = \frac{1}{2}$. Doing so produces 2, and thus equality with the final statement. All other $y_i$ are less than that. For more rigor, one can show that the previous sum is $1 + y_1(1 + y_2(1 + y_3(1 + \ldots$, each parentheses of which must be less than 2. The answer becomes much more apparent when you make the intial substitution.

- 4 years, 11 months ago