Dasdsad

Calculus - I

M.Thamban Nair
Department of Mathematics Indian Institute of engineering Madras

June 2006/Aug-Dec. 2010/Aug-Dec.2011

Contents

Preface 1 Sequence and series of Real numbers pool 1.1 M.T. Nair

vi 1 1 2 7 9 11 16 19 21 21 23 26 27 30

Sequence of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 convergence and divergence . . . . . . . . . . . . . . . . . . . Monotonic sequences . . . . . . . . . . . . . . . . . . . . . . . Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . Further examples . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy sequence . . . . . . . . . . . . . . . . . . . . . . . . . supernumerary exercises . . . . . . . . . . . . . . . . . . . . . . .

1.2

Series of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 Convergence and divergence of series . . . . . . . . . . . . . . both(prenominal) tests for convergence . . . . . . . . . . . . . . . . . . . Alternating series . . . . . . . . . . . . . . . . . . . . . . . . . Absolute convergence . . . . . . . . . . . . . . . . . . . . . . Additional exercises . . . . . . . . . . . . . . . . . . . . . . .

2 bourne, Continuity and Di?erentiability of Functions 2.1

M.T.



Nair 32 32 32 33 35 36 39

edge of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 2.1.2 Limit point of a set D ? R . . . . . . . . . . . . . . . . . . . Limit of a function f (x) as x approaches a . . . . . . . . . . Limit of a function in terms of a sequences . . . . . . . . . . 2.1.3 2.1.4 Some properties . . . . . . . . . . . . . . . . . . . . . . . . . Left limit and right limit ii . . . . . . . . . . . . . . . . . . . .

Contents 2.1.5 2.1.6 2.2 Limit at ? and at ?? . . . . . . . . . . . . . . . . . . . . . Additional Exercises . . . . . . . . . . . . . . . . . . . . . . .

threesome 40 42 43 43 46 51 52 52 53 54 55 59 61 61 62 63 64 66 70 71 73

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