lim [U(f, Pn) – L(f, Pn)] = 0, and in this case Sf = lim,+. U (f, Pn) = lim,-+ L(f, Pn). (b) For each n, let Pn be the partition of [0, 1] into n equal subintervals. Find formulas for U(f, Pn) and L(f, Pn) if f(x) = x. The formula 1+ 2+ 3+ .·+n = n(n + 1)/2 will be useful. ... (c) Use the sequential criterion for integrability from (a) to show directly that f(x) = x is integrable on [0, 1] and compute f, f.

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.