SAJC Revision Worksheet on Sequences & Series

1 Revision Worksheet on Sequences and Series A. (Sigma) notation Given n r r m u   , First term = mu Last term = nu No. of terms = 1n m  B. Basic properties/rules of summation No. Property Express the following examples in a similar form to the property given (the final answer is not required) 1 ( 1) n r m k n m k     , where m n Note: r is the index and k is treated as a constant 1n m  is the number of terms in the series 10 3 4 8 4 r   (Note : can use GC to check the answer if the limits are finite known values) 2 1 1 n n r r r r ku k u     8 8 8 1 1 1 5 1 1 5 5 r r r r r r                       3   1 1 1 n n n r r r r r r r u v u v           1 1 1 2 2 n n n r r r r r r r         4 1 1 1 where n n m r r r r m r r u u u m n          13 13 4 5 1 1 ! ! ! r r r r r r        5 Given the formula for nS , i.e. 1 n r r u   , the thn term of the series, 1 1 1 1 n n n r r r r n n u u u S S           Given 36 2 n nS   , 1 1 1 3 36 6 2 2 3 3 2 2 6 3 3 2 2 2 n n n n n n n n n n u S S                        r: index of sum m: lower limit n: upper limit Note: Lower limit of a sum does not always need to be 1.

2 C. Formula for some standard series No. Formula Example Remark 1   1 1 2 n r n nr   20 1 4 r r   20 1 4 r r    (20)(21)4 2  840 Using result of 1 1 n n r r r r ku k u     2   2 1 1 2 1 6 n r n n nr    Formula will be given in the question when required   1 (3 1) n r r    2 1 3 n r r r    2 1 1 3 n n r r r r      ( 1)(2 1) ( 1)3 6 2 n n n n n        ( 1) (2 1) 1 2 n n n   2( 1)n n  Using result of 1 1 n n r r r r ku k u     3 2 3 1 ( 1) 2 n r n nr      Formula will be given in the question when required 15 3 6 ( 2)r  15 15 3 6 6 2 r r r      15 5 15 3 3 1 1 6 2 r r r r r            2 2 (15)(16) (5)(6) (15 6 1)(2)2 2               14400 225 20   14195 change lower limit to 1 D. Limit and Convergence of Series Let 1 2 3 1 ... n n r n r S u u u u u        . When we write lim nn S S   , where S is a unique finite value , it means that nS approaches the finite value S as n   .  If lim nn S  = S, where S is a finite value, the series nS converges.  If lim nn S  does not exist or is infinite, the series nS diverges.

3 E. Recurrence Relations Recurrence relations are when the nth term is expressed in terms of the previous terms. Eg. 1 3n nu u  ; 1 3; u  . To find the terms, we do the following: When n =1, 2 1 3 3(3) 9u u   When n =2, 3 2 3 3(9) 27u u   When n =3, 4 3 3 3(27) 81u u   …and so on. Do remember how to use the graphing calculator to find the nth term of a recurre