Is there any relation between the definite integral and infinite series excluding the fact that an improper integral can be viewed as an infinite series whose terms are definite integrals? By definition, the definite integral is somewhat similar to an infinite series as it is a sum of infinite amount of terms.
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4$\begingroup$ In measure theory, sums may be interpreted as integrals over the integers with respect to a discrete measure; whereas integrals as we commonly think of them are with respect to the so-called Lebesgue measure. $\endgroup$– tghydeMay 2, 2013 at 13:39
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Yes, as thyde641 points out, infinite series (or, more generally, any sums) can be viewed as integrals if one knows about the more general idea of integration with respect to a measure. If you want further details, look up "measure theory".
