According to the definitions of "sum" and "equals" that most of us are used to, it makes no sense to say that 1 − 2 + 3 − 4 +... equals anything.
A generalized definition of the "sum" of a divergent series is called a summation method or summability method, which sums some subset of all possible series.
But if one examines the matter carefully, it turns out that the definition of the sum of all human beings is not the same as the species, for the sum of all human beings is not itself a rational, mortal animal.
As a result, we can achieve a set of rates such that (sum _{i=1}^{3}{R_{i}} geq frac {1}{2} C_{text {sum}}), and by the definition of normalized sum rate, we achieve (alpha = frac {1}{2}). 2 A cut Ω is a subset of (mathcal {V}) such that S∈Ω,D∉Ω, and (Omega ^{c} = mathcal {V} setminus Omega ). 3 The result for m=1 is trivial since we basically have an interference-free network.
We use the definition of the sum of probabilities to combine all the final probabilities, Eq. (2), for each objective separately.
From the definition of fractional sum the equation in the statement of the theorem is equivalent to frac{1-alpha}{B alpha)}g(t)+frac{alpha}{B alpha)}bigl( nabla _{a}^{-alpha}gbigr) (t)=f(t).
Applying the preceding inequality to every finite subset J of I gives the Bessel inequality :\sum_{i \in I}|\langle x, f_i \rangle|^2 \le \|x\|^2, \quad x \in H (according to the definition of the sum of an arbitrary family of non-negative real numbers).
This will correspond to the definition of the sum score approach.
So, from the definition of Gauss sums and the properties of a complete residue system mod p α and trigonometric sums, we have ∑ a = 1 p α χ ( a ) e ( a p α ) = ∑ r = 0 p − 1 ∑ a = 1 p α − 1 χ ( r p α − 1 + a ) e ( r p α − 1 + a p α ) = ∑ a = 1 p α − 1 χ ( a ) e ( a p α ) ∑ r = 0 p − 1 e ( r p ) = 0.
As we noted above, it follows that we cannot apply the Cauchy definition of infinite sums to the points of the line, and so happily we cannot immediately conclude that because they all have zero length so does the whole line.
In the mentioned work, the authors used an infinite sum to give a definition of discrete fractional sum, whereas Gray and Zhang used a finite sum in [20].
