# Interesting problem

## Main Question or Discussion Point

we have to prove that

10+100+1000+10000+100000+.........=-(1/9)

any ideas?

maybe you are trying to "prove" that
10+100+1000+10000+100000+.........=-(10/9)

here is a "proof" :)

let
S = 10 + 100 + 1000 + 10000 + ...

then
10S = 100 + 1000 + 10000 + ...

now,
S - 10S = 10
=> -9S = 10
=> S = -(10/9)
=> 10 + 100 + 1000 + 10000 + ... = -(10/9)

i dunno maybe this is what i actually was looking for.

but note that what i have given as a "proof" is not really a proof at all. the series 10 + 100 + 1000 + ... doesn't converge. so my "proof" doesn't actually work.

but note that what i have given as a "proof" is not really a proof at all. the series 10 + 100 + 1000 + ... doesn't converge. so my "proof" doesn't actually work.
so Murshid_islam what is the deal here? I can see that the series does not converge, however where is the problem on your proof? Is there a mathematical error, cause i could not see it, or what can we say about this?

The error was when he subtracted the two and got a fixed real number. Subtracting infinity from infinity is not a well-defined operation.

arildno
Homework Helper
Gold Member
Dearly Missed
Because, in his expression for 10S, he ignored the largest term present there.

as long as we are talking for infinit large numbers i cannot grasp how could there be a larger number on 10S than on S.I think it is absurd to talk about a "largest"term here, as long as we deal with infinit large terms! however i do understand the error now. SO defenitely we can say that
10+100+1000+10000+100000+.........=-(10/9)

is not mathematically true, and i cannot count on it, right?

Last edited:
HallsofIvy
Homework Helper
I would have thought it obvious from the start that a sum of positive numbers cannot be negative!

Yes, I recommend that you not count on it!

why don't you use sum of infinite G.P?

What does G.P mean at first place? I am sorry i am not used to these, so i really don't know what they stand for?
can you tell me?

I would have thought it obvious from the start that a sum of positive numbers cannot be negative!

!
Yeah, i also thought it could not be negative. However i saw this on a tv scientific show, and a proffesor demonstrated this, so i just wondered how that would be possible. That proffesor, whose name i cannot remember, said that he had turned this for a mathematical test to prove that this is right. If ,at first place, this is exactly what i saw, couse i am not 100% posotive.

What does G.P mean at first place? I am sorry i am not used to these, so i really don't know what they stand for?
can you tell me?
G.P - Geometric Progression

It is a series in which each term, apart from the first, is a fixed multiple of the previous term.

a + ar + ar^2 + ar^3 + ...+ar^n+...

The sum of the first n terms of such a series is a(1-rn)/(1-r). Check what happens for your series, when n tends to infinity.

thnx, i do know what a geometric progression is, but just did not know that g.p stands for that.
thnx indeed.

i think after we find the sum of that geometric progression using a(1-rn)/(1-r), and if we evaluate the limit of the result, it turns out that the sum must be infinity. Is that right?

i think after we find the sum of that geometric progression using a(1-rn)/(1-r), and if we evaluate the limit of the result, it turns out that the sum must be infinity. Is that right?
Yes, it diverges.

But, as mentioned earlier, the thing that should first convince you that the statement is not true is that the a sum of positive numbers cannot give you a negative number.

yeah, thank you guys for your help.

so Murshid_islam what is the deal here? I can see that the series does not converge, however where is the problem on your proof? Is there a mathematical error, cause i could not see it, or what can we say about this?
the error was when i let S = 10 + 100 + 1000 + .....
As the series doesn't converge i cannot let it equal to a number S.

Funny coincidence, but this Friday a prof at m university is giving a talk on why 1+2+4+6+...=-1 in the 2-adic numbers.