If the series is summable in closed form then clearly a rational function " ? " with this property exists; in fact it must always be a polynomial, and an upper bound on its degree can be found.
42.
When " X " & thinsp; is complete, a family " a " is unconditionally summable in " X " & thinsp; if and only if the finite sums satisfy the latter Cauchy net condition.
43.
In other words, if converges in probability to " X " sufficiently quickly ( i . e . the above sequence of tail probabilities is summable for all ), then also converges almost surely to " X ".
44.
The series is Ces�ro-summable in the weakest sense, called while requires a stronger form of Ces�ro's theorem, being Since all forms of Ces�ro's theorem are linear and stable, the values of the sums are as we have calculated.
45.
Often, a theorem is called " abelian " if it shows that some summation method gives the usual sum for convergent series, and is called " tauberian " if it gives conditions for a series summable by some method to be summable in the usual sense.
46.
Often, a theorem is called " abelian " if it shows that some summation method gives the usual sum for convergent series, and is called " tauberian " if it gives conditions for a series summable by some method to be summable in the usual sense.
47.
If a series is ( C, " k " ) ( Ces�ro ) summable for any " k " then it is Lambert summable to the same value, and if a series is Lambert summable then it is Abel summable to the same value.
48.
If a series is ( C, " k " ) ( Ces�ro ) summable for any " k " then it is Lambert summable to the same value, and if a series is Lambert summable then it is Abel summable to the same value.
49.
If a series is ( C, " k " ) ( Ces�ro ) summable for any " k " then it is Lambert summable to the same value, and if a series is Lambert summable then it is Abel summable to the same value.
50.
If a series is ( C, " k " ) ( Ces�ro ) summable for any " k " then it is Lambert summable to the same value, and if a series is Lambert summable then it is Abel summable to the same value.
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