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Unit IM — Series

Infinite Series °

Y. Introduction, 2, Sequences, Series.
onvergence 4. Ge
terms 6 Comparison tests, 7. Integra test. 8,Cmparsa fran DA cas

fonvergence of Expunental, Logarithmic and
sfr convergence. 17. Uniform convergence
formly convergent series. 20. Objectice Type of

Binomial series. 16. Procedure for testing
18. Weierstrass's Most. 19. Properties of y
Question.

91. INTRODUCTION

Infinite series occur so frequently in all types of problems thatthe necessity ofstudying their
convergence or divergence is very important. Unless a series employed in an investigation is
convergent, it may lead to absurd conclusions, Hence it is essential that the students of
engineering begin by acquiring an intelligent grasp ofthis subject.

92. SEQUENCES
(1) An ordered set of real numbers ay, 6 ay,

called a sequence and is denoted by
id t be an infinite sequence and

For instance (2) 1, 8,6, 7, (20 =D, sn (6) 1,1/2, 1/3, ny 1/8 «y
UP

(2) Limit. A sequence is said to tend t a lis
such that [a,~Z] <e for n 2N.

We then write Lt (¢,)=1 or simply (¢,) 1 a8 n +.

infinite sequences,
Tor every > 0, a value N ofn can be found

(8) Convergence. Ifa sequence (a) has a init limi is called a convergent sequence.
11 (,)is nat converbent it is said to be divergent.

In the above examples, (i) is convergent, while) and (i) are divergent,

(4) Bounded sequence. A sequence (,) in said t be bounded, her exist a number à
such that a, <A for every n,

(6) Monotonic sequence. The sequence (a)
steadily according a8 dy 12 dy OF dy 41, for all values on, Both incas
Sequences are called monotonic sequences

A monotonic sequence always tends to a limit, finite or infinite, Thus, a sequence which is
‘ronotonie and bounded is convergent.

said to increase steadily or to decrease
and decreasing

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