Get A Course in Advanced Calculus PDF

By Robert S. Borden

ISBN-10: 0486672905

ISBN-13: 9780486672908

An outstanding undergraduate textual content examines units and buildings, restrict and continuity in En, degree and integration, differentiable mappings, sequences and sequence, functions of wrong integrals, extra. difficulties. advice and strategies for chosen difficulties.

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Extra resources for A Course in Advanced Calculus

Example text

6. In Z+, define m

Assumethe inequality holds for somefl>-2. Then, sincex ) - l, 1* x > 0, and we can write nx)(l*x) -l+(n + l)x t nx2>l+(n + l)x. Note that equality holds if n - I or D x:0. (", txrl. * x,). r soME BASrCTNEQUALITTES 49 PRooF:If x,: x2=''' : xn, equality obviouslyholds. Now supposethe numbers are not all equal and that they have been arranged so that xr(rz increasingthe inequality of the theorem is equivalent to the inequality 1 [ t " , + . . 1,= ln x is concavedown, so the graph of y = ln x, x, g y:< rp, togetherwith the line segmentjoining the points (x,,ln x,) and (x,,ln xn), is comprisedby the boundary of a convex set in the xy plane; call this set S.

Analogously,if {x,} is decreasing,{x,} convergesto inf{xr}. x,(rr, i < j * x,1xi, then {x,} strictly increasing,ffid we write {x"} t. If i < j + x,2 xi, therr {xr} is a decreasingsequence,and we write {xr} :r. If i < j - x,) xi, then {x,} is strictly decreasing,ild we write {x"} J. A monotoni sequenceis either an increasingsequenceor a decreasingone; strictly monotone refers to the strictly increasing or strictly decreasing sequences. Then S has an accumulationpoint p. 38 CHAPTER2 LIMIT AND CONTINUITY IN E' pRooF: SinceS is bounded,it is containedin someinterval la,blCR.

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A Course in Advanced Calculus by Robert S. Borden

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