We math geeks like to play. And one thing we like to play with a lot are series. A series is an infinite row of numbers following a certain rule, all added up. Let me show you two examples:

S₁ = 1 + 1 + 1 + 1 + 1 + 1 + 1 + …

S₂ = 1 + 1/2 + 1/4 + 1/8 + 1/16 + …

The first series does NOT add up to a finite limit. (aka goes to infinity). The second DOES add up to a finite limit, 2. That means the second series is convergent, while the first series is divergent.

Convergence vs. Divergence

Here’s the interesting thing. When a series is divergent, we can do almost nothing useful with it. But when a series is convergent, we can do TONS and TONS of cool stuff with it!

In fact, one big branch of mathematics is simply devoted to figuring out which series are convergent and which are divergent. We often don’t even care WHAT the series converges to. It could be 1, or pi, or thirty-seven thousand, but it still converges. And we have a bunch of really clever tests for that.

But today, I won’t focus on that. Instead, I’ll show you a quick example of why fooling around with DIVERGENT series can leads to tons of trouble.

Let:
S = 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + …
From this, we obviously see that S = (1 – 1) + (1 – 1) + (1 – 1) + …, so S = 0 + 0 +0 + …, which means S = 0.
But, also:
S = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + …
And no matter how many zeros you tack on, you won’t change the initial number. So S = 1.

And because S = S, therefore 0 = 1.

Wait, there’s more!

How about this:
S = 1 – 1 + 1 – 1 + 1 – 1 + …
S = 1 – (1 – 1 + 1 – 1 + 1 – 1 + …)
S = 1 – S
2S = 1
S = 1/2
Yay!
So 0 = 1 = 1/2! And here I was all along, thinking they’re all different numbers…

Can you see the problem in the above calculations?

That’s the thing. The calculations above are absolutely fine! It’s the ORIGINAL PREMISE (that you can use divergent series in equations) that was wrong. It’s like using infinity in equations and then wondering why you end up with things like 0 = 1.

So from today’s article, I would like you to take away two things:

Firstly, remember to be VERY careful when you meet a divergent series.

Secondly, remember the three ways of summing up S above. They’re fantastic for confusing inexperienced mathematicians

1. richieacc Says:
   February 23rd, 2009 at 2:42 pm

   I don’t understand how you get the third equation, step 2 specifically. Would it not be:

   S = 1 – 1 + (1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + …)

   ?

   Which would evaluate to:
   S = 1 – 1 + S
   S = S

   ???
   I’m quite confused here.

2. Vlad Says:
   February 23rd, 2009 at 3:05 pm

   @richieacc:
   Let me try to explain in a bit more detail:

   S = 1 – 1 + 1 – 1 + 1 – 1 + …
   S = 1 + (-1 + 1 – 1 + 1 – 1 + …)

   Now throw a minus sign in front of the bracket.

   S = 1 – (1 – 1 + 1 – 1 + 1 – …)

   But the thing inside the bracket is now exactly the same as the original series! And so:

   S = 1 – S

   Does it make any more sense now? Let me know if you still don’t get it.

3. Dr Morrison Says:
   February 24th, 2009 at 3:16 pm

   To say divergent series are almost useless is wrong – they have great use in applied mathematics as asymptotic expansions, or to provide numerical approximations via Borel summation etc.

4. richieacc Says:
   February 24th, 2009 at 5:32 pm

   Thanks Vlad, makes sense now.

   Can’t explain why it made no sense earlier

5. Why is 0.99999… = 1? | Leeds Math Geeks Says:
   March 17th, 2009 at 11:48 am

   [...] course, whenever you hear a fancy proof like this, you have to be very careful. Like I showed in Why convergence matters, you can sometimes use seemingly impeccable logic to show complete [...]

6. Amy Says:
   June 3rd, 2010 at 11:51 pm

   To say divergent series are almost useless is wrong – they have great use in applied mathematics as asymptotic expansions, or to provide numerical approximations via Borel summation etc.

7. Chi Zhang Says:
   August 10th, 2010 at 5:00 am

   S=1-1+1-1+
   S=(1+1+1+1…)+(-1-1-1…)
   you can not add pos infinity and a neg infinity together

8. Vlad Says:
   August 10th, 2010 at 2:24 pm

   @Chi:

   I have a series for you to think about:

   1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + …

   Now, clearly, this series converges. But what happens if you take only the positive terms, and only the negative terms?

   (If you can’t figure it out, try googling “harmonic series”)

9. Chi Says:
   August 12th, 2010 at 7:56 am

   @Vlad:
   Sum could be 0 and 1 but never be 1/2
   2S=2-2+2-2+2…
   That will not equal to 1
   Try to sketch S = 1 – 1 + 1 – 1 + 1 – 1 + …
   It will show you that does not converge

   If you do this
   S = 1 – 1 + 1 – 1 + 1 – 1 + …
   S = 1 – (1 – 1 + 1 – 1 + 1 – 1 + …)
   it show that you may think first S is equal to 1
   the second S is equal to 0
   or the first is 0
   the second is 1

   you are a really geek
   thanks!

10. Vlad Says:
    August 12th, 2010 at 10:00 am

    @Chi:

    You’re absolutely right. 1+ 1 – 1 + 1 – 1 + 1… will never add up to 1/2. Neither will it add up to 1 or 0 because it’s a divergent series and so doesn’t have a sum.

    My last comment was only to point out the logical fallacy in your previous comment. Namely, the series…

    1 + 1/2 – 1/3 + 1/4 – 1/5 + 1/6…

    converges. Yet taking all the odd terms, they form a divergent series, and so do all the even terms. That doesn’t mean the full series itself is like adding two infinities .

    Your conclusion may have been right, but your argument had a flaw, and I couldn’t let you get away with that

11. Chi Says:
    August 13th, 2010 at 1:26 am

    I know what you want to say.
    The sum of two divergent series may be divergent or may not.

12. Vlad Says:
    August 21st, 2010 at 1:52 pm

    @Chi: Exactly

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