There is something more sophisiticated with summing of divergent series.

Well, you may have an idea, to what a series is summable - but then you have to make sure, it makes sense in all other relations.

One important concept is that of partial sums.

One other important concept was, to see the series in question either as a power-series, where powers of x are cofactored with the coefficients like

s(x) = 1 - 1x + 1x^2 + 1x^3 - ... +

or

t(x) = 1^x - 2^x + 3^x - 4^x + ... - ...

and then considering the appropriate limit of the occuring expression using the concept of partial sums.

Note, that the notation 1 - 1 + 1 - 1 + ... - ... can occur as limit of very different powerseries and thus can be assigned an arbitrary value.

The standard case is assuming lim x->1 s(x) or lim x->0 t(x).

The interesting aspect is, that the limits in those definitions are the same.

lim x->1 s(x) is formally equal to the powerseries lim x->1 1/(1+x); and the latter can be analytically continued beyond the radius of convergence. But if , for instance,

1 - 1 + 1 - ... + ... is seen as limit of

u(x) = (1-x^2)/(1-x^3) = 1 - 1 x^2 + 1 x^3 - 1 x^5 + 1 x^6 - ... + ...

then the limit for lim x->1 u(x) = 2/3, so it would be that also 1 - 1 + 1 - 1 + ... - ... = 2/3

(see example in K.Knopp "Theorie und Anwendung unendlicher Reihen" (there is an english version available), chap XIII)

It may be misleading, that series like this without reference of their context are given as examples in books or online-media at all... I think, that, if there is no explicite reference is given, we have the (still ambiguous) "default", to interpret the series either as coefficients of a powerseries with consecutive powers (without holes) or as a zeta-series (as far as possible).

The series with which we deal in tetration are not automatically power- or exponential-series; for instance the Lambert-W, or the integrals involve coefficients with k^k, where k is the series-index. So we must be extra careful with the assignement of values to iterative expressions like lim k->inf a^^k or the like.

The problems of the ambiguity of 0^0 when seen as lim x->0 0^x or seen as lim y->0 y^0 are well known, and it is well known, that they don't converge to the same value (which would be necessary for a unique definition).

Gottfried

Well, you may have an idea, to what a series is summable - but then you have to make sure, it makes sense in all other relations.

One important concept is that of partial sums.

One other important concept was, to see the series in question either as a power-series, where powers of x are cofactored with the coefficients like

s(x) = 1 - 1x + 1x^2 + 1x^3 - ... +

or

t(x) = 1^x - 2^x + 3^x - 4^x + ... - ...

and then considering the appropriate limit of the occuring expression using the concept of partial sums.

Note, that the notation 1 - 1 + 1 - 1 + ... - ... can occur as limit of very different powerseries and thus can be assigned an arbitrary value.

The standard case is assuming lim x->1 s(x) or lim x->0 t(x).

The interesting aspect is, that the limits in those definitions are the same.

lim x->1 s(x) is formally equal to the powerseries lim x->1 1/(1+x); and the latter can be analytically continued beyond the radius of convergence. But if , for instance,

1 - 1 + 1 - ... + ... is seen as limit of

u(x) = (1-x^2)/(1-x^3) = 1 - 1 x^2 + 1 x^3 - 1 x^5 + 1 x^6 - ... + ...

then the limit for lim x->1 u(x) = 2/3, so it would be that also 1 - 1 + 1 - 1 + ... - ... = 2/3

(see example in K.Knopp "Theorie und Anwendung unendlicher Reihen" (there is an english version available), chap XIII)

It may be misleading, that series like this without reference of their context are given as examples in books or online-media at all... I think, that, if there is no explicite reference is given, we have the (still ambiguous) "default", to interpret the series either as coefficients of a powerseries with consecutive powers (without holes) or as a zeta-series (as far as possible).

The series with which we deal in tetration are not automatically power- or exponential-series; for instance the Lambert-W, or the integrals involve coefficients with k^k, where k is the series-index. So we must be extra careful with the assignement of values to iterative expressions like lim k->inf a^^k or the like.

The problems of the ambiguity of 0^0 when seen as lim x->0 0^x or seen as lim y->0 y^0 are well known, and it is well known, that they don't converge to the same value (which would be necessary for a unique definition).

Gottfried

Gottfried Helms, Kassel