1

Let $a_i$ be a sequence of real numbers and let p>1. Then:

$\sum_{i \in\mathbb{N}} |a_i|\leq (\sum_{i \in\mathbb{N}} |a_i|^p)^{1/p}$.

"Proof": $\sum_{i \in\mathbb{N}} |a_i|=((\sum_{i \in\mathbb{N}} |a_i|)^p)^{1/p}\leq (\sum_{i \in\mathbb{N}} |a_i|^p)^{1/p}$

where the last inequality holds by Jensen's inequality. This proof works for the case n=2, a case in which it clearly must be false.

Annie
  • 307
  • 1
  • 7

1 Answers1

0

You seem to be using $$\left(\sum_{i=1}^\infty|a_i|\right)^p\leq\sum_{i=1}^\infty|a_i|^p.$$ But this inequality holds in the opposite direction.

Gerhard S.
  • 1,248
  • I know that this inequality has to actually hold in the opposite direction, but doesn't Jensens inequality state that this direction is actually true? – Annie Jan 28 '19 at 18:29
  • Jensen's inequality says that the value of a convex function at a convex combination of several arguments does not exceed the convex combination of the function values at those arguments. Where is the convex combination in your inequality? – Gerhard S. Jan 28 '19 at 18:57
  • The reason why Jensen can't be applied is that it only holds for probability spaces. – Annie Jan 29 '19 at 15:40