Advanced Calculus:Suppose that f is an integrable function on [a,b]. Suppose that for each n, Sn is a Riemann sum for f corresponding to a partition of width < 1/n. Prove that The image shows a mathematical expression related to calculus, specifically the concept of definite integrals. The expression is:\[\lim_{{n \to \infty}} S_n = \int_{a}^{b} f(x) \, dx.\]This equation represents the relationship between a limit of a sequence of partial sums \(S_n\) and the definite integral of a function \(f(x)\) over the interval \([a, b]\). It signifies that as the number of partitions \(n\) approaches infinity, the Riemann sum \((S_n)\) approaches the exact area under the curve of \(f(x)\), which is given by the integral from \(a\) to \(b\).

Given f is an integrable function on a,b. Therefore for any partition Pn, such that Pn→0 it satisfies: limn→∞Uf,Pn-∫abfxdx=0 and limn→∞Lf,Pn-∫abfxdx=0 The given Riemann sum of the given function is Sn corresponding to the partition Pn with width <1n. Hence Pn<1n→0. Since Sn is Riemann sum, therefore: Lf,Pn≤Sn≤Uf,Pn.Therefore: Lf,Pn-∫abfxdx≤Sn-∫abfxdx≤Uf,Pn-∫abfxdxlimn→∞Lf,Pn-∫abfxdx≤limn→∞Sn-∫abfxdx≤limn→∞Uf,Pn-∫abfxdx0≤limn→∞Sn-∫abfxdx≤0 Hence limn→∞Sn-∫abfxdx=0 Which implies that limn→∞Sn=∫abfxdx.

Answered: lim Sn | f(x) dæx. a ||

Math

Advanced Math

lim Sn | f(x) dæx. a ||

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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