Cauchy Schwarz?

Calculus Level 3

Let x 1 , x 2 , … , x n x_1, x_2, \ldots , x_n be non-zero reals such that ∣ x 1 ∣ + ∣ x 2 ∣ + ⋯ + ∣ x n ∣ = 1 |x_1 | + |x_2 | +\cdots + |x_n | = 1 .

Determine which of the following equation/inequation must be true.

n 2 ≤ ∑ k = 1 n 1 ∣ x k ∣ n^2 \leq \sum_{k=1}^n \frac 1{\mid x_k\mid } n 2 = ∑ k = 1 n 1 ∣ x k ∣ n^2 = \sum_{k=1}^n \frac 1{\mid x_k\mid } n 2 ≥ ∑ k = 1 n 1 ∣ x k ∣ n^2 \geq \sum_{k=1}^n \frac 1{\mid x_k\mid }

Mohammad Hamdar by Mohammad Hamdar , 22, Lebanon

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Mohammad Hamdar
Mar 28, 2017

We have n 2 = ( ∑ k = 1 n 1 ) 2 = ( ∑ k = 1 n ∣ x k ∣ . 1 ∣ x k ∣ ) 2 { n }^{ 2 }={ (\sum _{ k=1 }^{ n }{ 1 } ) }^{ 2 }={ (\sum _{ k=1 }^{ n }{ \sqrt { \left| { x }_{ k } \right| } .\frac { 1 }{ \sqrt { \left| { x }_{ k } \right| } } ) } }^{ 2 } ≤ ∑ k = 1 n ( ∣ x k ∣ ) 2 ∑ k = 1 n 1 ( ∣ x k ∣ ) 2 ( B y C a u c h y − S c h w a r z i n e q u a l i t y ) \le \sum _{ k=1 }^{ n }{ { (\sqrt { \left| { x }_{ k } \right| } ) }^{ 2 } } \sum _{ k=1 }^{ n }{ \frac { 1 }{ { (\sqrt { \left| { x }_{ k } \right| } ) }^{ 2 } } }\quad (By\quad Cauchy-Schwarz \quad inequality) = ∑ k = 1 n ∣ x k ∣ ∑ k = 1 n 1 ∣ x k ∣ = ∑ k = 1 n 1 ∣ x k ∣ =\sum _{ k=1 }^{ n }{ \left| { x }_{ k } \right| } \sum _{ k=1 }^{ n }{ \frac { 1 }{ \left| { x }_{ k } \right| } } =\sum _{ k=1 }^{ n }{ \frac { 1 }{ \left| { x }_{ k } \right| } }

2 Helpful 0 Interesting 0 Brilliant 0 Confused
J Joseph
Mar 27, 2017

if all xi are the same and look like 1/y the sum is n 2 n^{2}

if they aren't all the same but still look like 1/y then the sum is greater than n 2 n^{2} Didn't consider the rest

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...