Sum of an infinite geometric series

I just finished reading the .99... Equals or does not equal 1 debate from 2012. What a fascinating read! Well at least for the first half. Being a self proclaimed math hobbiest ( passed high school level math only) I was drawn to ydr11's arguments that .33.. does not accurately represent 1/3 as it is unresolved. Not only do the 3's go on forever but so does the remainder of 1 that causes the next 3,and so on and so on, leaving it in a perpetually unresolved state.

Anyhow I have been doing some more reading trying to understand this issue and found that at the core of my resistance to the mainstream position is that finding a finite value for the sum of an infinite geometric series appears impossible. This assumption that it is possible, seems to be at the root of all the proofs that show .99... Equals 1.

Consider the series 0.3+ 0.03+ 0.003... Most will agree that it is an infinite series i.e. it has no last term, and continues forever. Most will also agree that there is no term n for which the sum of the series equals exactly 1/3, except for n=infinity. This is where I have the issue. I know that "when" is not a definable math term but If someone could give me a general idea of when it adds up to exactly 1/3 that would help me tremendously.

Is it before or after the terms continue on forever?

Before would be impossible as it would not then be an infinite series

After seems equally impossible as there is no last term or "after"

Some may say it is neither, it is more of while it's in an infinite state, but that seems to be the same as the before position and implies that it was not In fact infinite, or that the total could go above 1/3.

Perhaps the question cannot be answered because there is no when? If there is no when, how did it happen?

Disclaimer: I actually haven't read that thread. However, I am a skilled mathmaticians who can help you through these questions. Have you taken calculus (or have a good precal class)? The answers to your questions relates to understanding convergence and limits.

No I have not, from what I have read it sounds like convergence heads towards zero, 1/2 + 1/4 + .... And Divergence heads towards infinity 9 + 90 + 900...? Limits I do not know. Are they the numbers that the series heads towards? In the example above the limit is 1/3?

Quote: 1call2many

Limits I do not know. Are they the numbers that the series heads towards? In the example above the limit is 1/3?

Yeah, limits are what something goes to as something else goes to infinity. In this case, .3+.03+.003+...->1/3, as the number of terms goes to infinity the sum goes to 1/3. The limit gives us that for any tiny number, no matter how small, there exists a number terms that if you add them up the difference between the limit (1/3) and the sum of those terms will be less than that tiny number. Using that definition it makes sense to say that infinite number of terms sums to the limit.

Wikipedia's limit article is good as of my last reading of it

Quote: 1call2many

No I have not, from what I have read it sounds like convergence heads towards zero, 1/2 + 1/4 + .... And Divergence heads towards infinity 9 + 90 + 900...? Limits I do not know. Are they the numbers that the series heads towards? In the example above the limit is 1/3?

That actually isn't true. For a convergent series the terms have to tend to zero but that is not enough to guarantee the series converges. For instance the series 1/1+1/2+1/3+... diverges as the sum tends to infinity. If you are interested I wrote a post awhile ago about geometric series and you can check it out at http://www.bubblews.com/news/2407002-9999991 . Hopefully it is pretty straight forward but if you have any questions just ask and I'll try to answer.

[. The limit gives us that for any tiny number, no matter how small, there exists a number terms that if you add them up the difference between the limit (1/3) and the sum of those terms will be less than that tiny number. Using that definition it makes sense to say that infinite number of terms sums to the limit.

Wikipedia's limit article is good as of my last reading of it

It sounds like you are saying what I thought I already knew that the further you care to travel along the series, the tinier the difference between sum and limit? Or no matter how small a number we think of we can add more terms to get a little closer and hence a smaller difference?

That says to me we will never reach the limit. ( how could you reach anything with no end to the addition). I notice the Wikipedia link kept using the term arbitrarily close, is arbitrarily close, close enough to say let's just roll with it, it makes all our calculus work, etc.etc. There must be more to it, mathematitions like to be precise, arbitrarily close and acts like it's there, does not seem like a compelling reason to believe that it can sum to the limit.

On a side note how do you quote someone hear and get that nice grey box? My attempt does not look like that.

Quote: 1call2many

On a side note how do you quote someone hear and get that nice grey box? My attempt does not look like that.

There's a quote button at the bottom right of each post. It prefills the reply box with some code and you enter your response after it.

Quote: 1call2many

It sounds like you are saying what I thought I already knew that the further you care to travel along the series, the tinier the difference between sum and limit? Or no matter how small a number we think of we can add more terms to get a little closer and hence a smaller difference?

Both, really. No matter how small a number we can think of, once you've added enough terms, you'll not only be closer, but never be further again, no matter how many more terms you add. For that to be true, the difference has to keep getting smaller and smaller as you add more terms.

Quote: 1call2many

That says to me we will never reach the limit. ( how could you reach anything with no end to the addition). I notice the Wikipedia link kept using the term arbitrarily close, is arbitrarily close, close enough to say let's just roll with it, it makes all our calculus work, etc.etc. There must be more to it, mathematitions like to be precise, arbitrarily close and acts like it's there, does not seem like a compelling reason to believe that it can sum to the limit.

It does so by definition. An infinite sum is a kind of limit, and a repeating decimal is a kind of infinite sum. Since that's what's meant in the first place, it's proper to talk about them being what they're defined to be.

Quote: 1call2many

It sounds like you are saying what I thought I already knew that the further you care to travel along the series, the tinier the difference between sum and limit? Or no matter how small a number we think of we can add more terms to get a little closer and hence a smaller difference?

That's a good intuitive start.

Quote: 1call2many

That says to me we will never reach the limit. ( how could you reach anything with no end to the addition). I notice the Wikipedia link kept using the term arbitrarily close, is arbitrarily close, close enough to say let's just roll with it, it makes all our calculus work, etc.etc.

At infinity we have reached the limit, however, using English for these discussions is problematic because it isn't specific enough. Don't think of adding the terms ones at a time. Imagine they are weights already sitting on a scale and you are trying to find a weight which counterbalances the other side.

Quote: 1call2many

There must be more to it, mathematitions like to be precise, arbitrarily close and acts like it's there, does not seem like a compelling reason to believe that it can sum to the limit.

We do like to be precise. Infinity is a tricky thing, intuition often breaks down out there. Sadly, for math like this you either need to "take our word for it" or be willing to suffer through many semesters of college math so you can prove it yourself.