limit of a sequence - Limit of a Sequence Advanced Level...

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Limit of a Sequence Advanced Level Pure Mathematics Advanced Level Pure Mathematics Calculus I Chapter 2 Limit of a Sequence 2.1 Introduction 2 2.2 Sequences 2 2.3 Convergent Sequences 6 2.4 Divergent Sequences and Oscillating Sequences 7 2.5 Operations on Limits of Sequences 8 2.6 Sandwich Theorem for Sequences 13 2.7 Monotonic Sequences 17 2.8 The Number e 23 2.9 Some Worked Examples 24 2.1 INTRODUCTION Prepared by. K. F. Ngai Page 1 2
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Limit of a Sequence Advanced Level Pure Mathematics Some examples of sequences: 1. 1, 2 1 , 4 1 , 8 1 , , 1 2 1 - r , 2. 1, - 1, 1, - 1, , ( - 1) r+1 , 3. cos x , cos2 x , cos3 x , , cos rx , 2.2 SEQUENCES Definition 2.1 A sequence { x n } is a function on the set of real numbers, and is usually written as x 1 , x 2 , x 3 , , x n , . 1. The term x n is called the general term of the sequence. Example {1, 2, 4, 8, . ..} is a sequence of positive integers with general term 1 2 - = n n x . 2. If the sequence has infinite number of terms, it is called an infinite sequence. 3. If the sequence has finite number of terms, it is called a finite sequence. 4. S n = x 1 + x 2 + x 3 + + x n + is said to form a series . 5. S n = = n r r x 1 = x 1 + x 2 + x 3 + + x n is a finite series . 6. S n = = 1 r r x = x 1 + x 2 + x 3 + + x n + is an infinite series . How to find the series sum , = = n r r n x S 1 ? By using (M1) Mathematical induction . (M2) Method of difference where ) ( ) 1 ( r f r f x r - + = such that ) 1 ( ) 1 ( ] ) ( ) 1 ( [ 1 1 f n f r f r f x S n r n r r n - + = - + = = = = (M3) Partial fractions and method of difference . (M4) Standard formulae : (i) A.P. : a , a + d , a +2 d , , a +( n - 1) d , . Prepared by. K. F. Ngai Page 2
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Limit of a Sequence Advanced Level Pure Mathematics S n = ] ) 1 ( 2 [ 2 1 d n a - + . (ii) G.P. : a , a r , a r 2 , , a r n - 1 , . S n = r r a n - - 1 ) 1 ( ; S = r a - 1 where | r | < 1 . (iii) 1 + 2 + 3 + + n = = n r r 1 = ) 1 ( 2 1 + n n 1 2 + 2 2 + 3 2 + + n 2 = = n r r 1 2 = ) 1 2 )( 1 ( 6 1 + + n n n 1 3 + 2 3 + 3 3 + + n 3 = = n r r 1 3 = 2 2 ) 1 ( 4 1 + n n Example 1 If x n denotes the n th term of the series which begins ... ) 3 2 1 )( 2 2 1 ( 1 3 ) 2 2 1 ( 1 2 1 1 2 2 2 2 2 2 + + + + + + Prove that the sum of the series to n terms is ). (3 2 1 2 n x n - Example 2 Show that (sin2 θ +sin4 + +sin2 n ) sin = sin n sin ( n +1) . Example 3 (a) Find = + n r r r 1 ) 1 ( 1 . (b) If u r = r ( r +1)( r +2) for positive integer r , show that 3 r ( r +1) = u r - u r - 1 . Prepared by. K. F. Ngai Page 3
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Limit of a Sequence Advanced Level Pure Mathematics Hence, or otherwise, find = + n r r r 1 ) 1 ( . (M5)
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limit of a sequence - Limit of a Sequence Advanced Level...

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