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Césaro's method

We will discuss a simple one here, not the more complicated Levin's u-transform that Maple actually uses in its evalf(Sum(...)) construct. Note also that in the share library, there is a whole package of such sequence acceleration techniques. See ?share and ?share,trans for details.



is called the Césaro limit of . Note that it is a limit of average values of . Many theoretical results are known about this method (see Hardy's book Divergent Series, for example), including the fact that it is regular. This means that if converges in the ordinary sense, then the Césaro limit converges to the same thing.

Weniger remarks that regularity is actually not what we really want, because it seems that regular methods won't speed convergence up very much; Levin's u-transform is not a regular method, for example. But regularity is a nice theoretical property, anyway, and the example is rather simple.

Example. If , then if n is even and if n is odd. This means that

We can check that for the first few values of n and see that we have made no blunders (unless I have a typo up there).

Then the average value of up to n is

Clearly this tends to as , so the Césaro limit of is . Incidentally, this is what evalf(Sum( (-1)^k, k=0..infinity)) gets, so the Levin's u-transform method agrees with the Césaro sum, which is also the same (in this case) as putting x=-1 in .

Let us code this in Maple.

cesaro := proc( a::algebraic, n::name)
  local m,k, s, avg;
  s := sum( a, n=0..k); # Hardwired assumption
  avg := sum(s, k=0..m)/(m+1);
  limit( avg, m=infinity)

This works symbolically, and not numerically, but I think it is clear that we could do the same thing numerically, by computing , , and so on up to some predetermined number, and then computing the running averages and seeing if that settles down.

After defining this procedure, cesaro((-1)^k, k) gives as expected. More interestingly, cesaro( sin(k*theta), k) gives .

Indeed the Fourier sine series for is exactly

(you can do the calculations yourself), but the series on the right does not converge in the usual sense. By the way, evalf(Sum) fails on this one, for numerical values of .

Break for a stretch

next up previous
Next: 2nd Half Up: Applied Math 475/563b Notes Previous: Convergence acceleration in

Robert Corless
Fri Jan 30 13:19:12 EST 1998