What is Divergent integrals: Definition and 14 Discussions

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series














{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.}
The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme.
In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A summability method or summation method is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent series




{\displaystyle 1-1+1-1+\cdots }
the value 1/2. Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization.

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  1. A

    A Extending reals with logarithm of zero

    What do you guys have to say about this Mathoverflow post? Do you have any interesting ideas about this? https://mathoverflow.net/questions/432396/extending-reals-with-logarithm-of-zero-properties-and-reference-request
  2. A

    A Multiplying divergent integrals using Hardy fields approach

    I wonder if the following makes sense. Suppose we want to multiply ##\int_0^\infty e^x dx\cdot\int_0^\infty e^x dx##. The partial sums of these improper integrals are ##\int_0^x e^x dx=e^x-1##. Now we multiply the germs at infinity of these partial sums: ##(e^x-1)(e^x-1)=-2 e^x+e^{2 x}+1##...
  3. A

    A What physical meaning can the “determinant” of a divergency have?

    I am [working][1] on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\det w=\exp(\operatorname{reg }\ln w)$$ which is analogous to how determinant of a matrix can be...
  4. A

    A Why physicists cannot renormalize all divergent integrals?

    Why the physicists have troubles with infinities in many physical theories, such as quantum gravity? Why cannot they just use divergent integrals and regularize or renormalize them in the end so to obtain finite values? I mean, operations on divergent integrals are not a problem, and techniques...
  5. A

    A Algebra of divergent integrals

    Hello, guys! I would like to know your opinion and discuss this extension of real numbers: https://mathoverflow.net/questions/115743/an-algebra-of-integrals/342651#342651 In essence, it extends real numbers with entities that correspond to divergent integrals and series. By adding the rules...
  6. Ramtin123

    A Computing amplitude for divergent loop diagrams?

    I am trying to compute the cross-section for the diagram below with a divergent triangle loop: where ##X^0## and ##X^-## are some fermions with zero and negative charge respectively. I am interested in low energy limits, so you can consider W-propagator as ##\frac {i\eta_{\mu\nu}} {M_w^2}##...
  7. saybrook1

    Determining Cauchy principal value of divergent integrals

    Homework Statement So I've found a ton of examples that show you how to find cauchy principal values of convergent integrals because it is just equal to the value of that integral and you prove that the semi-circle contribution goes to zero. However, I need to find some Cauchy principal values...
  8. Sinisa

    Where is sum of divergent series used in physics?

    Hello everyone! My question is: Where is sum of divergent series and divergent integrals used in physics? What it all means? Where can I find examples of divergent integrals? Is there a book of problems for physicists? I am mathematician. I developed a method for summing divergent series...
  9. Z

    Zeta regularization of divergent integrals

    From the model used in the zeta regularization procedure to give a meaning to divergent series in the form 1+2+3+4+... , we propose a similar method to give a finite meaning to divergent integrals in the form \int_{0}^{\infty}dx x^{m} for positive 'm' in terms of the negative values of the...
  10. A

    Convergent and Divergent Integrals

    I had a question regarding convergent and divergent integrals. I want to know the "exact" definition of an improper integral that converges. Wikipedia states that For a while, I took that as a valid answer and claimed that any integral that has a finite answer must be convergent. However, I...
  11. Z

    Cauchy trick for divergent integrals.

    is this trick valid at least in the 'regularization' sense ?? for example \int_{-\infty}^{\infty} \frac{dx}{x^{2}-a^{2}} then we replace thi integral above by \int_{-\infty}^{\infty} \frac{dx}{x^{2}+ie-a^{2}} for 'e' tending to 0 using Cauchy residue theorem i get...
  12. M

    Resummation of divergent integrals.

    if we can obtain resummation methods for divergent series such as 1-1+1-1+1-1+1-1+... or 1!-2!+3!-4!+.. my question is why is there no method to deal with divergent integrals like \int_{0}^{\infty} dx x^{s-1} or \int_{0}^{\infty} dx (x+1)^{-1} (x^{3}+x)
  13. K

    Renormalization and divergent integrals.

    If the problem of renormalization is that there are divergent integrals for x-->oo couldn't we make the change. \int_{0}^{\infty}dx f(x) \approx \sum_{n=0}^{\infty}f(nj) using rectangles with base 'j' small , and approximating the divergent integral by a divergent series and 'summing' by...
  14. K

    How Can Divergent Integrals Be Handled with Ramanujan Summation?

    Are there any method to deal with divergent integrals in the form \int_{0}^{\infty}dx \frac{x^{3}}{x+1} \int_{0}^{\infty}dx \frac{x}{(x+1)^{1/2}} ? in the same sense there are methods to give finite results to divergent series as 1+2+3+4+5+6+7+... or 1-4+9-16+25 or similar