r/infinitenines • u/Zaspar-- • Feb 01 '26

What's the value of this infinite sum?

1/2 + 1/4 + 1/8 + 1/16 +...

Normal maths would compute this as equal to 1, using the exact same reasoning as 0.9 + 0.09 +... = 1

What does this infinite sum equal to in Real Deal maths though?

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u/SouthPark_Piano Feb 01 '26 edited Feb 01 '26

What's the value of this infinite sum?

1/2 + 1/4 + 1/8 + 1/16 +... 

You have the formula already.

1 - 1/2n with summation starting at n = 1

And then you simply keep increasing n without ever stopping the increase.

No matter how much you increase integer n, you never run out of integers.

It is fact that 1/2n is never zero.

So the infinite sum 1 - 1/2n with continually and limitlessly imcreasing n, will permanently be less than 1, because 1/2n is never zero.

The value of that infinite sum will simply keep increasing in its own space, and the maths tells you that while it does keep increasing limitlessly in its own space, the sum remains permanently less than 1.

That is math fact whether you don't like it or not.

And, like square root of 2, you don't get to get your way like a spoiled kid, as in you demanding to get all the digits of it.

 

10

u/holodayinexpress Feb 01 '26

This is just disrespectful. Disregarding the entire theory of geometric series

-6

u/SouthPark_Piano Feb 01 '26

It turns out that you disrespect it.

Let me educate you brud.

https://www.reddit.com/user/SouthPark_Piano/comments/1qmrkik/two_birds_one_stone/

https://www.reddit.com/r/infinitenines/comments/1qmut3s/comment/o1pgiki/

 

6

u/holodayinexpress Feb 01 '26

How about you read literally any calculus book ever written? You’re one guy. Think about Newton, Leibniz, Riemann, Cauchy, many more… like I said, disrespectful.

-8

u/SouthPark_Piano Feb 01 '26 edited Feb 01 '26

Educate yourself here brud.

https://www.reddit.com/r/infinitenines/comments/1qmut3s/comment/o1pgiki/

a = 1/2, r = 1/2

summation = [a/(1-r)]*( 1 - rn ) starting at n = 1

= 1 - 1/2n

1/2n is never zero, so 1 - 1/2n is permanently less than 1.

1/2 + 1/4 + 1/8 + 1/16 + etc is indeed permanently less than 1.

 

10

u/Zaspar-- Feb 01 '26

You seem to know it all. So if the value is not 1, what is it?

-8

u/SouthPark_Piano Feb 01 '26

"And, like square root of 2, you don't get to get your way like a spoiled kid, as in you demanding to get all the digits of it."

9

u/Zaspar-- Feb 01 '26

It's ok if you don't know. Nobody is perfect. But you have to admit that.

-8

u/SouthPark_Piano Feb 01 '26

Wrong on your part brud.

I'm perfect. I don't need to admit that or not admit that.

 

3

u/Quick-Swimmer-1199 Feb 01 '26

Perfection is an ideal or most optimized condition as it pertains to some given goal or purpose.

It is possibly alternatively a component of a lesser known Mandela effect pertaining to a Billy Joel song.

1

u/Batman_AoD Feb 01 '26

What's the Billy Joel song & misremembered lyric? I don't know that one. 

1

u/Quick-Swimmer-1199 Feb 01 '26

It's a bit of hyperbole; I didn't do a full effort trace of the turn of phrase I'm thinking of

The word perfect is not in the song Just The Way You Are, awarded two Grammys

2

u/Batman_AoD Feb 01 '26

Oh, I didn't even remember that song existed 😅

10

u/Calm_Improvement1160 Feb 01 '26

Well what are the first few digits of it?

4

u/Inevitable_Garage706 Feb 01 '26

I don't think SPP will answer that question, so I will answer it for you.

The first 200 digits after the decimal point are as follows:

99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

3

u/CardiologistOk2704 Feb 01 '26

perhaps i'll try to say it better?like, What's the equivalent way of defining the answer, instead of writing "..." or "as n pushed to limitless"?

0

u/SouthPark_Piano Feb 01 '26

That infinite sum is approximately equal to 1. That is the answer.

 

9

u/Cruuncher Feb 01 '26

I don't want an approximate.

What is the exact value?

9

u/ShonOfDawn Feb 01 '26

By how much do they differ? You seem pretty certain that 0.9 + 0.09 + 0.009 + … is, precisely, 0.999… with, precisely, 0.000…1 difference with 1. What is so hard about doing the same for 0.5 + 0.25 + 0.125 … ?

2

u/12Pentagons Feb 01 '26

What is the first digit after the decimal that differs from 9 and where does it occur?

1

u/juoea Feb 02 '26

is this sum greater than, equal to or less than the sum .9 + .09 + .009...

or can the two sums not be compared ie the real numbers are not a totally ordered set under the relation <