Infinite infinities!

Level 1

What is the sum of the following series?

S =(1 + 1/2 + 1/4 + 1/8 .....) + (1/2 + 1/4 + 1/8 + 1/16...) + (1/4 + 1/8 + 1/16 + 1/32....)......

The answer is 4.

1 solution

Rindell Mabunga
Dec 21, 2013

First let's recall the sum of infinite progression

$S = {a_1}/{1 - r}$

where:

S is the sum of the Arithmetic Progression
$a_1$ is the first term of the Arithmetic Progression
r is the common ratio between terms

Let $S_1$ be $1 + 1/2 + 1/4 + ...$

$S_2$ be $1/2 + 1/4 + 1/8 + ...$

$S_3$ be $1/4 + 1/8 + 1/16 + ...$

$...$

such that $S = S_1 + S_2 + S_3 + ...$

Using the formula, we substitute the values of the unknowns on each sum

$S_1 = 1/{1 - 1/2} = 1/{1/2} = 2$

$S_2 = {1/2}/{1 - 1/2} = {1/2}/{1/2} = 1$

$S_3 = (1/4}/{1 - 1/2} = {1/4}/{1/2} = 1/2$

$...$

Therefore

$S = S_1 + S_2 + S_3 + ... = 2 + 1 + 1/2 + ... = 2/{1 - 1/2} = 2/{1/2} =$ 4

OMG! I did the exactly same thing until get S1 = 2 , S2 = 1 and S3 = 0.5 ........but end up giving up because 3.5 is not an integer. After seeing your comment, I recheck and found that I never see the "....." after the last bracket.

With thanks. Besides, here's a little suggestion with no offense at all , please bracket up the number orders like in your conclusion. It take me several tries and few of time to figure out why 2/1 - 1/2 = 4 but not 1.5 (its like 2-0.5), as well as why 2/1/2 (seems like 2.5) is equal to 4.

Pigeon Gymnastic - 7 years, 3 months ago

Infinite infinities!

The answer is 4.

1 solution

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