Sum ideas (part 2)

So, what about divergent series?

A divergent series has no limit, so we can’t assign that as a value. Conventionally we just say it has no value, its value is undefined. But in fact some divergent series can be associated with a definite quantity by means other than the limit of the sequence of partial sums. For some purposes, it can be useful to regard that quantity as the value, or a value, of the series.

There are lots of ways to associate a value with a divergent series; lots of summation methods, as they are called. It’s unfortunate terminology, in that it suggests they’re methods for finding the unfindable sum of the infinite number of numbers. But it’s the terminology we’re stuck with.

Summation methods are kind of like technical standards: the great thing about them is there’s so many to choose from. Generally a given summation method can be used with some series, but not with others. Some methods are stronger than others, in the sense that the one can be applied to any series the other can, with the same result, but it can also be applied to some series the other can’t handle.

Perhaps the simplest summation method applicable to a divergent series is Cesàro summation. In its simplest form, this is just finding the limit not of the partial sums of a series, but the average of the partial sums. For example, Grandi’s series is

1 - 1 + 1 - 1 + 1 - 1 + ....

The first partial sum is 1, the second is 1-1=0, the third is 1-1+1=1, the fourth is 1-1+1-1=0, and so on — they alternate between 1 and 0. They don’t converge. But the average of the first one partial sum is 1, the average of the first two is 1/2, the average of the first three is 2/3, the average of the first four is 1/2, and so on, forming the sequence

\displaystyle 1, \frac{1}{2}, \frac{2}{3}, \frac{1}{2}, \frac{3}{5}, \frac{1}{2}, \frac{4}{7}, \frac{1}{2}...

and that sequence does converge, to 1/2. This value is the Cesàro summation of Grandi’s series.

Now, we know if we add together a finite number of integers, we get an integer, and it seems crazy to think you could sit down and add an infinite number of integers and get a fraction. Then again, it’s crazy to think you could sit down and add an infinite number of integers. And that’s not what we’re doing. But we know the partial sums alternate between 0 and 1, so the value halfway between, 1/2, in some sense does characterize the behavior of the infinite series.

A reasonable question to ask is, what’s the Cesàro summation of a convergent series? It doesn’t take too much thinking to realize intuitively that if a series converges, then the average of the partial sums also should converge and to the same value. For instance, the partial sums of

S = \displaystyle \sum_{k=0}^{\infty} \frac{1}{k!}

converge to e, and so do the averages of the partial sums. Granted, the partial sums converge much faster: after just 12 terms the partial sum is 2.718281828... while after 10000 terms the average of the partial sums is still 2.717281831.... But it’s getting there. A summation method that gives the conventional limit when applied to a convergent series is called regular. Cesàro summation is regular, and clearly that’s a nice attribute to have: it means Cesàro summation is consistent with ordinary summation, but is stronger in the sense that it also gives results for some series which have no classical value.

But does it make sense to say

\displaystyle 1 - 1 + 1 - 1 + 1 - 1 + ... = \frac{1}{2}?

It’s an unconventional and probably misleading use of the equal sign, but if it’s understood you’re talking about a value assigned using a summation method, specifically Cesàro summation, you maybe can get away with it. But you can also make the sense more explicit. Hardy again:

We shall make systematic use of the following notations. If we define the sum of \Sigma a_n, in some new sense, say the ‘Pickwickian’ sense, as s, we shall say that \Sigma a_n is summable (P), call s the P sum of \Sigma a_n, and write

\Sigma a_n = s (P).

We shall also say that s is the P limit of the partial sum s_n, and write

s_n \to s (P).

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