# Why is this trollmath wrong?

• Jul 12th 2013, 06:47 AM
SpringFan25
Why is this trollmath wrong?
Hi forum,

I saw this trollmath on youtube. i dont claim its correct, but i cant understand where the mistake is. help :)

Consider a geometric series as follows:

$S = 2 + 4+ 8 + 16 + ......$ [1]

$2S = 4 + 8+ 16 + ......$ [2]

[2]-[1]

$S = 2$

Clearly the last line is false, but where is the mistake?
• Jul 12th 2013, 07:01 AM
Plato
Re: Why is this trollmath wrong?
Quote:

Originally Posted by SpringFan25
I saw this trollmath on youtube. i dont claim its correct, but i cant understand where the mistake is. help Consider a geometric series as follows:

$S = 2 + 4+ 8 + 16 + ......$ [1]

$2S = 4 + 8+ 16 + ......$ [2]

[2]-[1]
$S = 2$ Clearly the last line is false, but where is the mistake?

You are only allowed to subtract finite sums. SO

$S_n = 2 + 4+ 8 + 16 + ...+2^n$ [1]

$2S_n = 4 + 8+ 16 + ....+2^{n+1}$ [2]

$2S_n-S_n=2^{n+1}-2$
• Jul 12th 2013, 07:46 AM
HallsofIvy
Re: Why is this trollmath wrong?
That is not exactly correct. You can subtract infinite sums if they converge. For example, 1+ 1/2+ 1/4+ 1/8+ ... (adding 1 over powers of 2) is a "geometric series" that converges to 1/(1- 1/2)= 2. Similarly, 1+ 1/3+ 1/9+ 1/27+ ... (adding 1 over powers of 3) is a geometric series that converges to 1/(1- 1/3)= 3/2. (1+ 1/2+ 1/4+ 1/9+...)- (1+ 1/3+ 1/9+ 1/27+ ...)= (1/2- 1/3)+ (1/4- 1/9)+ (1/8- 1/27)+ ...= 1/6+ 5/36+ 19/216+ ... converges to 2- 3/2= 1/2.

The problem here is that the series does not converge.
• Jul 12th 2013, 07:51 AM
Plato
Re: Why is this trollmath wrong?
Quote:

Originally Posted by HallsofIvy
That is not exactly correct. You can subtract infinite sums if they converge.

Of course that is true. But the point is that to prove convergence one looks at finite sums.
• Jul 12th 2013, 08:54 AM
Soroban
Re: Why is this trollmath wrong?
Hello, SpringFan25!

Quote:

Consider a geometric series as follows:

. . $\begin{array}{ccccc}S &=& 2 + 4 + 8 + 16 + \hdots & {\color{red}[1]} \\ 2S &=& \quad\;\; 4 + 8 + 16 + \hdots & {\color{red}[2]} \end{array}$

${\color{red}[2 ] - [1]}:\;S = -2$

Clearly the last line is false, but where is the mistake?

The mistake is in thinking that we can "do" arithmetic with Infinity.

Clearly: . $S \:=\:2 + 4 + 8 + 16 + \hdots \:=\:\infty$

Then: . $2S \:=\:\infty$

Hence: . $2S - S \:=\:\infty-\infty$, an indeterminate form.

That is, $S$ could be any value.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\text{Evaluate: }\:S \;=\;1 - 1 + 1 - 1 + 1 - 1 + \hdots$

Method 1

$S \;=\;(1-1) + (1-1) + (1-1) + \hdots$

$S \;=\;0 + 0 + 0 + \hdots$

$\boxed{S \;=\;0}$

Method 2

$S \;=\;1 - (1-1) - (1-1) - (1-1) - \hdots$

$S \;=\;1 - 0 - 0 - 0 - \hdots$

$\boxed{S \;=\;1}$

Method 3

$S$ is a geometric series
. . with first term $a = 1$ and common ratio $r = \text{-}1.$

Hence: . $S \;=\;\frac{a}{1-r} \;=\;\frac{1}{1-(\text{-}1)}$

Therefore: . $\boxed{S \;=\;\tfrac{1}{2}}$
• Jul 12th 2013, 10:54 PM
agentmulder
Re: Why is this trollmath wrong?
Quote:

Originally Posted by SpringFan25
Hi forum,

I saw this trollmath on youtube. i dont claim its correct, but i cant understand where the mistake is. help :)

Consider a geometric series as follows:

$S = 2 + 4+ 8 + 16 + ......$ [1]

$2S = 4 + 8+ 16 + ......$ [2]

[2]-[1]

$S = 2$

Clearly the last line is false, but where is the mistake?

We CAN get the right answer if we are careful

$2S = (2 + 2) + (4 + 4) + (8 + 8) + ....$
$\ \ S = \ \ \ \ \ 2 \ \ \ \ + \ \ \ \ 4 \ \ \ + \ \ \ \ 8 \ \ \ \ + ...$

Subtract vertically to get S

Just because we have the difference of two infinite divergent sums doesn't necessarily put the right answer out of reach , like Soroban said infinity minus infinity could be anything... that includes the right answer. Finding a procedure that produces the right answer may be difficult though.

:)