does this apply only for terms that are less than 1 eg 1 + 1/2 + 1/4…. or does this apply to all ap/gp. I remember studying this, but it’s been so long I remember only the gist.
In: 248
I’m a bit late here, but most of the answers seem to be missing the point.
The sums of some infinite series give finite values because we define them to do so. Which is how maths works.
The question we’re trying to ask is “what do we get when we add an infinite number of things together?”
We are asking what we even mean when we say “+ … = ”
Normal algebra and number theory cannot handle this. We cannot actually add together an infinite number of things. But geometry suggests there should be a way of doing this (we get Zeno’s paradoxes, and things like 1/2 + 1/4 + 1/8 +…). So we need to come up with a way! And the magic trick we come up with is to use **limits**.
At the risk of getting into the maths too much, if we have some sum-to-n terms (or “partial sums), Sn, we say our infinite sum S∞ converges to a limit L if:
> for any ε>0 there exists some N so large that n≥N implies that |Sn−L|<ε
Translating this from maths into real words, we are setting up a game here. You challenge me by giving me some ε – as small as you like (but bigger than 0), and I have to find some point in the sequence of partial sums (our Sn) for which *all subsequent terms* are within ε of the limit L.
Taking our classic 1/2 + 1/4 + 1/8 + … + 1/2^n + … = 1 result, Sn = 1/2 + 1/4 + … + 1/2^n
Let’s say you give me ε = 0.001. I have to find an N for which SN is somewhere between 0.999 and 1.001, and Sn will be within that rage for all subsequent terms.
In this case, with a bit of sneaky maths, I can pick N = 10.
> S10 = 1/2 + 1/4 + … + 1/2^10 = 0.9990234375
And we can show that Sn will always be within our target range for any larger n.
Generalising, it will turn out that given any ε, provided N > -ln(ε)/ln(2) we’re good.
We are basically saying that the limit of an infinite series or sequence exists if there is some value we can get arbitrarily close to, even if we never actually get there.
In practice we won’t use this formal definition for every sequence. Instead we’ll come up with all sorts of tests for convergence, which tend to rely on showing that our sequence is more converge-y than a sequence we know converges (like the one above), or less converge-y than one we know doesn’t converge.
————
With a bit of algebra we can see that the only **arithmetic progression** that converges will be the trivial one (0 + 0 + 0 + …). Our formula for the partial sum-to-n terms of an arithmetic series with starting number a and common difference d (so a + a+d + a+2d + a+3d + …) is
> Sn = n/2 [2a + (n − 1) × d ]
As n goes to infinity, this will only be nicely behaved if a = d = 0.
For a **geometric progression**, with starting number a and common ration r (so a + ar + ar^2 + ar^3 + …) we get a sum-to-n terms of:
> Sn = a(1-r^(n))/(1-r)
Which will converge nicely to a/(1-r) if |r| < 1
For other series we’d have to mess around with stuff to figure it out. Or just look it up.
————
But what about series which *don’t* converge? A classic example is:
> 1 – 1 + 1 – 1 + … = ?
What does it mean to say “+ … =” for a series that doesn’t converge?
That series above will have partial sums of either 1 or 0. So it cannot converge to a particular value. But what if we still want an answer?
For this example we can use a geometric series (where a = 1, r = -1). We’re not supposed to put this value into our sum-to-infinity term for geometric progressions, but what if we do?
> S∞ = a/(1-r) = 1/(1–1) = 1/2
Which… is surprisingly reasonable. Sure, our partial sums will never actually be 1/2, they’ll either be 1 or 0. But they’ll bounce between 1 and 0 forever, which averages out to 1/2.
This definition:
> 1 – 1 + 1 – 1 + … = 1/2
doesn’t make sense in terms of numbers or algebra, and it doesn’t even work in terms of limits (our new fancy maths), but on some level it does still make sense. So it might be useful, and is definitely interesting.
And if we explore this, getting into complex analysis, we can get some neat results, such as:
> 1 + 1 + 1 + 1 + … = -1/2
> 1 + 2 + 3 + 4 + … = -1/12
and so on.
These sums of infinite terms don’t make sense in normal number theory, or in algebra, or with limits. But that’s because using those areas of maths “+ … =” doesn’t make any sense for these series. We need a new definition and a new area of maths. A way of extending the idea of limits of infinite series to divergent series.
There are a whole bunch of ways of coming up with answers to the question “what does + … = mean for divergent series?” and what makes it really interesting is that for many of these series, and many methods, we end up getting the same results. Which suggests there is something meaningful to all of this.
Latest Answers