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  2. Divergence - Wikipedia

    en.wikipedia.org/wiki/Divergence

    In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source or a sink at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of the field vectors exiting from an infinitesimal region of space than entering it.

  3. Harmonic series (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Harmonic_series_(mathematics)

    One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two: + + + + + + + + + + + + + + + + + + Grouping equal terms shows that the second series diverges (because every grouping of convergent series is only convergent ...

  4. Convergent series - Wikipedia

    en.wikipedia.org/wiki/Convergent_series

    The Maclaurin series of the logarithm function ⁡ (+) is conditionally convergent for x = 1 (see the Mercator series). The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.

  5. Vector calculus identities - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus_identities

    As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. The divergence of a tensor field of non-zero order k is written as ⁡ =, a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar.

  6. Divergent series - Wikipedia

    en.wikipedia.org/wiki/Divergent_series

    A summation method is regular if, whenever the sequence s converges to x, A(s) = x. Equivalently, the corresponding series-summation method evaluates A Σ (a) = x. Linearity. A is linear if it is a linear functional on the sequences where it is defined, so that A(k r + s) = k A(r) + A(s) for sequences r, s and a real or complex scalar k.

  7. Convergence tests - Wikipedia

    en.wikipedia.org/wiki/Convergence_tests

    If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]

  8. Divergence of the sum of the reciprocals of the primes

    en.wikipedia.org/wiki/Divergence_of_the_sum_of...

    For a positive integer x, let M x denote the set of those n in {1, 2, ..., x} which are not divisible by any prime greater than p k (or equivalently all n ≤ x which are a product of powers of primes p i ≤ p k). We will now derive an upper and a lower estimate for | M x |, the number of elements in M x.

  9. Divergence theorem - Wikipedia

    en.wikipedia.org/wiki/Divergence_theorem

    In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, [1] is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.