Let we have a continuous function $f(x)$ in the interval $ [ a,b ] $

Does there exist any relationship between its integral and summation of function-values defined at the integers between $a$ and $b$.

i-e Relationship between $\int_a^b f(x) dx$ and $\sum_{i= \lceil a\rceil}^{\lfloor b\rfloor} f(i)$ ?

For instance we have an integral test for infinite series which if positive and decreasing, then both integral and summation converges. But what can be inferred about the partial sum of series (not necessarily decreasing) if we know the integral between some finite limits ?

  • 2
    $\begingroup$ We can say useful things if $f$ is monotone. $\endgroup$ – André Nicolas Sep 22 '14 at 5:58
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    $\begingroup$ In general: nothing. Knowing the average of a function over an interval tells you nothing about its value at the endpoints (unless we impose conditions such as increasing/decreasing). $\endgroup$ – Winther Sep 22 '14 at 5:58
  • $\begingroup$ However, putting some weights before the $f(i)$ can provide you the integral of the functions $f$. This can be made exact for polynomials en.wikipedia.org/wiki/Numerical_integration $\endgroup$ – Quickbeam2k1 Sep 22 '14 at 6:00
  • $\begingroup$ @AndréNicolas So, if the function is monotone then? $\endgroup$ – kaka Sep 22 '14 at 6:02

Yes. Euler-McLaurin's formulas completely describe this kind of relations.


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