Sum ideas (part 3)

Another summation method is Abel summation. The idea is, if we have

\displaystyle S = \sum_{k=0}^{\infty} a_k

then we can define

\displaystyle f(r) = \sum_{k=0}^{\infty} a_kr^{k}

for r \in [0, 1). Then if \lim_{r\to 1^-} f(r) exists and is finite, that limit is the Abel sum of S.

As a simple example, apply this to Grandi’s series, 1 - 1 + 1 - 1 .... Here f(r) = 1 - r + r^2 - r^3 ... = 1/(1+r) for |r|<1. The limit as r\to 1 is 1/2, the same result as we obtained using Cesàro summation. In fact it can be shown that Abel summation is stronger than Cesàro summation, i.e., for series that can be Cesàro summed, Abel summation give the same result, but there are additional series which can be Abel summed but not Cesàro summed. Of course Cesàro summation is consistent with ordinary summation for convergent series, and therefore so is Abel summation: that is, Abel summation is regular.

Here’s another example. Consider

\displaystyle 1-2+3-4+... =  \sum_{k=0}^{\infty} (-1)^k(k+1).

Not only does this series not converge, but the partial sum averages don’t converge either, so it is not Cesàro summable. But it is Abel summable. We make this sum into a function

\displaystyle f(r) =  \sum_{k=0}^{\infty} (-1)^k(k+1)r^k = r^0-2r^1+3r^2-4r^5...

But now notice: r^0 = -d(-r)^1/dr, -2r^1 = -d(-r)^2/dr, 3r^2 = -d(-r)^3/dr, and so on:

\displaystyle f(r) =  -\frac{d}{dr}\sum_{k=0}^{\infty} (-r)^k

and, again, for |r|<1,

\displaystyle f(r) =  -\frac{d}{dr}\frac{1}{1+r} = \frac{1}{(1+r)^2}.

Now the Abel sum is \lim_{r\to1^-} f(r) = 1/4.

A couple more properties (besides regularity) a summation method might have are linearity and stability. For the following let \mathbf {A}(r) denote the result of applying summation method \mathbf {A} to series r. By linearity is meant: if \mathbf {A}(\sum a_n) = s and \mathbf {A}(\sum b_n) = t then \mathbf {A}(\sum ka_n+b_n) = ks+t. By stability is meant: if \mathbf {A}(a_0+a_1+a_2+...) = s then \mathbf {A}(a_1+a_2+...) = s-a_0, and conversely. Cesàro summation and Abel summation both are linear and stable. So is classical summation, for that matter.

You can prove that for any linear and stable summation method \mathbf{A}, the sum of the Grandi series \mathbf{A}(g) is 1/2, if that sum exists:

\mathbf{A}(g) = \mathbf{A}(1-1+1-1+1...) = 1 + \mathbf{A}(-1+1-1+1...) (by stability)
= 1 - \mathbf{A}(1-1+1-1...) (by linearity)
= 1 - \mathbf{A}(g) and so
\mathbf{A}(g) = 1/2.

That “if that sum exists” provision is important. For instance, classical summation of the Grandi series is undefined, not 1/2, even though classical summation is linear and stable. You can come up with similar proofs about linear and stable sums of other series, that they must always have some particular value if they have a value at all. Showing that they do indeed have a value is another matter!

Conversely, you can prove some series do not have values for any summation method that is linear and/or stable. For example, suppose \mathbf{A} is stable and \mathbf{A}(\sum_{k=0}^\infty 1) has value a. Then

a = \mathbf{A}(1+1+1+1+1...) = 1 + \mathbf{A}(1+1+1+1...) (by stability)
a = 1 + a,

an impossibility. So \sum_{k=0}^\infty 1 cannot be summed by any stable summation method. There are unstable methods, however, that can sum that series.

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