Answer to Question #200129 in Real Analysis for Muhammad tayyab

Question

Show that the 2^n is not convergent.

Answer 1

Solution:

We know that a necessary and sufficient condition for the convergence of a sequence is that it is bounded and has a unique limit point.

$2^{n}=(1+1)^{n} \geq 1+n, \forall n \in \mathbb{N}$

If P>0 is any real number howsoever large, we have,

1+n>P, whenever n>P-1 .

Let m be a positive integer, such that, m>P-1.

$\Rightarrow$ For any real number P>0, $\exists$ a positive integer, m, such that $2^{n}>P, \forall n \geq m$

$\Rightarrow$ The sequence $\left\{2^{n}\right\}$ is not bounded.

$\Rightarrow$ The sequence $\left\{2^{n}\right\}$ diverges to $\infty$ .

$\Rightarrow$ The sequence $\left\{2^{n}\right\}$ is not convergent.

Hence, proved.