Titlebook: Real Analysis; Foundations and Func Miklós Laczkovich,Vera T. Sós Textbook 2015 Springer New York 2015 Fourier series.Stieltjes integral.co

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Functions of Bounded Variation,ce between either sum and the integral is at most ., the oscillatory sum corresponding to ...Thus the oscillatory sum is an upper bound for the difference between the approximating sums and the integral..We also know that if . is integrable, then the oscillating sum can become smaller than any fixed

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The Improper Integral,nts of the interval) and are bounded on that interval. These restrictions are sometimes too strict; there are problems whose solutions require us to integrate functions on unbounded intervals, or that themselves might not be bounded.

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The Definite Integral,This concept, in contrast to that of the indefinite integral, assigns numbers to functions (and not a family of functions). In the next chapter, we will see that as the name . that they share indicates, there is a strong connection between the two concepts of integrals.

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Functions of Bounded Variation,ence between the approximating sums and the integral..We also know that if . is integrable, then the oscillating sum can become smaller than any fixed positive number for a sufficiently fine partition (see Theorem 14.23).

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Infinite Sequences II, that is, .. ≠ 0 for all . > .., then 1 ≤ .. ≤ 9, and so . also holds if . > ... By Theorem 4.17, .. Thus for a given . > 0, there is an .. such that . for all . > ... So if ., then ., and thus .. → 1.

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