Fandom

Math Wiki

Binomial inequality

1,069pages on
this wiki
Add New Page
Talk0 Share

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

The binomial inequality is a statement about raising a sum to a natural power:

\forall x \in \R, x \ge -1, and \forall n \in \N : (1 + x)^n \ge 1 + nx.

It is essentially a truncated version of the binomial theorem, that is easily proved by induction.

Proof

Check it's true for n = 1 : (1 + x) \ge 1 + x, clearly.

Now assume true for some n = k \in \N : (1 + x)^k \ge 1 + kx.

As  1 + x \ge 0, we can multiply that inequality through by (1 + x) and preserve the inequality:

 (1 + x)^{k+1} \ge (1 + x)(1 + kx) = 1 + x + kx + kx^2.

As kx^2 \ge 0 \forall x \in \R, \forall n \in \N : 1 + x + kx + kx^2 \ge 1 + (1 + k)x.

So, (1 + x)^{k+1} \ge 1 + (k + 1)x, so by induction it is true for all n \in \N.

Also on Fandom

Random Wiki