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  3. 4 pp 253 marsden amp hoffman investigate the validity of...

Question: 4 pp 253 marsden amp hoffman investigate the validity of...

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4. [pp. 253, Marsden & Hoffman] Investigate the validity of Theorem 5.3.1 for the sequence fn defined by fn (x) = 1 nx25.3.1 Theorem Suppose thatfi.s...are integrable functions on a closed bounded interval [a. b], and that they converge uniformly to a limit function f on [a, b]. Then f is integrable on [a, b] and limJn The idea is that for large n, the values f,(x) are all uniformly close to f(x). Thus, any Riemann sum for f is close to the corresponding Riemann sum for f. The finite length of the interval is important at this step. Since f, is integrable, its Riemann sums are all close together if the mesh of the partitions is small enough, and so those for f are also close together. In particular, the upper and lower sums for f are close together. With appropriate details, this shows that fis integrable. The assertion that the limit of the integrals is equal to the integral of the limits then follows from the observation that rb Selecting n large enough that lh(x)-f(x)| < є for all x E [a,b] makes this smaller than (b - a)e and gives us our assertion about limits. Again, the finite length of the interval is important. The details of the proof are supplied at the end of this chapter. If we apply this result to the partial sums of an infinite series of integrable functions that converges uniformly on a closed bounded interval [a,b], we find that we can interchange the order of integration and summation.

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