Question

Find the sum of the given infinite series.
$\displaystyle \sf \large \sum^{ \infty }_{n = 0} {2}^{n}$
​

Answer 1

Answer

The sum of the infinite series is divergent.

Reason

The mathematical notation $\displaystyle{\sum^{\infty}_{n=0} 2^n}$ stands for $1+2+4+8+...$, which is adding two times the previous number from 1 infinitely.

Intuitive Method

The sum of the squares' area, which side length ratios are all √2 is equal to the series sum. (Picture included)

Using the sum of different series

Since $2^n>n\log2$, we get the relation of two different series as

• $\displaystyle{\sum^{\infty}_{n=0} 2^n>\displaystyle{\sum^{\infty}_{n=0} n\log2}}$

But $\displaystyle{\sum^{\infty}_{n=0} n\log2}$ diverges to infinity, which is less than $\displaystyle{\sum^{\infty}_{n=0} 2^n}$. Therefore, the sum of the series diverges to infinity.

More information:

It is possible to evaluate the sum of the series in the following manner, using number properties.

• Commutative
• Associative
• Identity
• Distributive

Let's evaluate the given series.

$S=1+2+4+8+...$

$=1+2(1+2+4+8+...)$

$=1+2S$

$\therefore S=-1$

This method can generate many values for different series. Let's use commutative property on a series.

$S=1-1+2-2+4-4+8-8+...$

$=1+(2-1)+(4-2)+(8-4)+...$

$=1+1+2+4+...$

$=1+S$

$\therefore S$ is undefined.

But if we evaluate in a different method, we get another value.

$S=1-1+2-2+4-4+8-8+...$

$=(1-1)+(2-2)+(4-4)+(8-8)+...$

$=0$

$\therefore S=0$

We showed this method can give different values for one series. And since every term is a positive number in $\displaystyle{\sum^{\infty}_{n=0} 2^n}$, their sum cannot be a negative number.

Answer 2

$\sf{\displaystyle\sum _{n=0}^{\infty \:}2^n}$

$\bigstar\textbf{\textsf{ Apply Series Geometric Test}}$

$\mathrm{If\:the\:series\:is\:of\:the\:form\:}\sum _{n=0}^{\infty }r^n$

$\displaystyle \bullet\quad \mathrm{If\:}|r|<1\mathrm{,\:then\:the\:geometric\:series\:converges\:to\:}\frac{1}{1-r}$

$\displaystyle \bullet\quad \mathrm{If\:}|r|\ge \:1\mathrm{,\:then\:the\:geometric\:series\:diverges}$

$\sf{\mathsf{Here}\: r=2,\:|r|=2\ge 1,\:\mathsf{by\:the\:geometric\:test\:criteria}}$

$=\textbf{\textsf{Diverges}}$