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Convergence in Sequences and Series: Understanding Riemann Sums and Integral Tests

Explore how sequences and series converge using Riemann Sums, Integral Tests, and Remainder Estimates in mathematical analysis.

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Convergence in Sequences and Series: Understanding Riemann Sums and Integral Tests

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  1. Sequences and Series Sequences: All three sequences converge Series:

  2. Sequences and Series 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A sequence an is defined by the height of the nth bar, which is equal to the area of the nth rectangle. The series sn is the Riemann Sum up to n. The series converges if the area is finite as n

  3. The integral test f (x) a1 a2 a3 a4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 The sequence an can be drawn so that it lies entirely above the function f or entirely below it. Thus the convergence is the same for the seriesand the improper integral

  4. Remainder estimate using the integral f (x) an an+1 an an+1 n n+1 where (error in estimate of s from adding the first n terms)

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