EJC 2026 H3 Mathematics - Sequences and Series - Notes, Tutorial, and Solutions to Tutorial

1 H3 MATHEMATICS (9820) SEQUENCES AND SERIES Syllabus Content • Sequences and series, e.g. general terms, sum, limiting behaviours, bounds • Summation of series by the method of differences • Integration by Reduction formulae 1 Method of Differences Consider the sum [ ] 1 f( ) f( 1 ) n r rr = −−∑ for some function f. By writing it as an explicit sum, we can see that many terms cancel when evaluating this: [ ] 1 f( ) f( 1 ) [ f( 1 ) f( 0 ) f( 2 ) f( 1 ) f( 3 ) f( 2 ) f( 2 ) f( 3 ) f( 1 ) f( 2 ) f( ) f( 1 ) ] f( ) f( 0 ) n r rr nn nn nn n = −−= − +− +− + + −− − + −− − +−− = − ∑  This is called the method of differences because ru is a difference of two functions in r. Example 1 Find 3 1 1n r rr= −∑ using the method of differences.

2 2 Integration by Reduction Formulae Example 2 [2017/PJC/Prelim/Q3] Let 0 3 2 d1 n n xux x= + ⌠⌡ . (i) Evaluate, in exact form, (a) 01uu+ [3] (b) 012uuu−+ [2] (c) 2u [2] (ii) Deduce the value of 0u and 1u . [2] (iii) Show that 21 2n nnu u n +− − = . [3]

3 Example 3 If cos dn nI xx=∫ , show that 1 2 11sin cos n nnI nx nx In − − −= + for 2,n n +≥∈  . Hence find (a) 4cos dxx∫ and (b) 2 0 π 5cos dxx∫ .

4 3 Convergence of sequences and series In university, you will learn the epsilon-delta definitions of limits and convergence. Here, we will content ourselves with an intuitive understanding of convergence. Roughly speaking, A sequence 123,,,uu u  converges to a limit L if all later terms get arbitrarily close to L. We write this as lim n nuL →∞ = , or nuL→ as n→∞ . If there does not exist such a limit L, we say that the sequence diverges. Example: 1 2 k ku = does not converge to –1 because later terms do not get arbitrarily close to –1. In fact, for all values of k, ( 1) 1ku −− > . The sequence converges to 0 because 0ku − approaches 0 as k →∞ . A series 123uu u+++  converges to a limit L if the sequence of partial sums converges to L: That is, the sequence ( ) ( )1 12 123,, ,uuu uu u+ ++  converges to L We write this as 1 r r uL ∞ = =∑ , or 1 r n r uL = →∑ as n→∞ . Example: The series 1 1 2k i ∞ = ∑ refers to the sequence of partial sums 1 11 111, , ,...1 12 124+ ++ . Since that sequence converges to 2, we write 1 1 22k i ∞ = =∑ . Algebra of Limits Assume that () na and () nb are sequences such that () na converges to L and () nb converges to M. Then the following hold as n→∞ : (i) (addition) nn LMab → + + (ii) (subtraction) nn LMab → − − (iii) (scalar multiplication) nca cL→ for any constant c. (iv) (modulus) na L→ (v) (product) nnab LM→ (vi) (quotient) n n a b L M→ if 0M ≠ (vii) (powers) k n ka L→ for any constant k. The following results are useful tests for convergence of a sequence/series: The first result says that an increasing sequence bounded above is convergent. If a sequence 123,,,uu u  satisfies: • 123uu u≤≤≤  , and • there is a k such that nuk< for all n then the sequence is convergent.

5 The second result is a squeeze theorem for sequences: Given three sequences ( ), ( ), ( )nnnabc such that nnna bc≤≤ for each n, if both () na and () nc converge to a limit L then () nb also converges to L. The third result is a squeeze theorem for series: Given three sequences ( ), ( ), ( )nn nuvw such that nn nu vw≤≤ for each n, if both 1 r n r u = ∑ and 1 r n r w = ∑ are convergent then 1 r n r v = ∑ is also convergent. A corollary is the comparison test: Given two series ( ),( )nnab such that 0 nn b a≤≤ for each n: • If 1 r r a ∞ = ∑ diverges, then 1 r r b ∞ = ∑ also diverges. • If 1 r r b ∞ = ∑ converges, then 1 r r a ∞ = ∑ also converges. Two well-known results follow from these. You are expected to be familiar with both the results and their proofs. Example 4 Prove that the series 1 9 11 1 4 16+++ +  converges. Example 5 Prove that the harmonic series 111 35 11 2 4+++++  does not converge.

6 Example 6 [2020/RI/Prelim/Q1b] Let (1111. 2357 11) 1 n nS n=−+−− +− + (i) State the sum of 2 24 .1 ( 1) nnxx x+−−+−  [1] (ii) Hence show that 221 20 ( 1) d ,1 n n n xSA xx + = +− +∫ where A is a constant to be determined. [3] (iii) Deduce that 1 ,23 nSA n−< + and hence determine the limit of the sequence nS . [3]

7 Tutorial Questions 1. Find 1 2 2 4 83 n r rr= ++∑ in terms of n. 2. By applying the method of differences to the function f defined by 32f( ) 2 3n n nn=++ , show that ( )( ) 1 2 12 1 6 k r kk kr = ++=∑ . 3. It is given that 2 2 0 d 1 n n xI x x α − =⌠ ⌡ , 0 1α< < , 0n≥ . Show that 21 2 12( 1) (2 1 1) n nnnI nI αα+ ++= −+− . 4. Differentiate 1 sin cosn x x− with respect to x. Hence, given that 0 1 dcos t n nIx x=⌠⌡ , prove that 21 sin( 1) ( 2)cos nn n tnI n I t −−− = +− . 5. If 21 0 ednx nIx x −=∫ , prove that 1 22 ( 1) ennI nI − −= −− for 2n≥ and evaluate 1I . Hence or otherwise, find the exact value of 9I . By considering the magnitudes of nx and 2 e x− for 0 1 x≤≤ and 0n≥ , show that 01 nI<< and deduce that 65 65e24 22<< . 6. [2017/Q2] (i) Let y be a differentiable function of x. For any positive integer n, prove that ( ) 1 1 d dd d dd n nn n nn yyxy x nx xx − −= + [5] (ii) For any non-negative integer n, define ( ) 22 dee d n xx n ny x −= . (a) Find 0y , 1y and 2y . [3] (b) Prove that ( )21 2 21 0nn ny xy n y++ + ++ = , for 0n≥ . [4] (c) Hence prove that ( ) ( )1 d 21d nny nyx + = −+ , for 0n≥ . [3]

8 7. Determine whether each of the following sequences converge or diverge. (a) sin 2 n nu π= , 1, 2, 3,n= … (b) 3 3 n n nu = , 1, 2, 3,n= … (c) ! n n nu n= , 1, 2, 3,n= … (d) sin n nu n= , 1, 2, 3,n= … 8. [N2002/Q5 (S Paper)] The geometric progression U has terms 123,,,uu u …, with common ratio r , where | |1r < . It is given that 2 iivu= for 1,2,3,i= … . (i) Show that 2 1 11 1 nn ii ii uvu r= = = +∑∑ . It is given further that 1i iiwvv += − for 1,2,3,i= … . (ii) Show that 2 1 11 (1 ) nn ii ii wu r u = = = −∑∑ . Let 1 Ui i Su ∞ = =∑ , 1 Vi i Sv ∞ = =∑ and 1 Wi i Sw ∞ = =∑ . Show that (iii) 1 12u VU S SS u+= , (iv) 2 1WSu = . 9. (i) Sketch the graph of sinyx= and determine the largest constant a and the smallest constant b such that sin bxax x≤ ≤ for 0. 2x π≤≤ (ii) Hence determine for every positive integer k, if the series 111sin sin sin ...123 kk k +++ is convergent or divergent.

9 10. The thr term ru of an arithmetic progression U is given by ( 1) , for 1, 2, 3, .ru ar d r=+− = … The progression is such that there exists a term of U equal to 2a . Given that a and d are positive integers, show that (a) for each term ru of U , there exists a term of U equal to 2 ru , (b) for each term ru of U , there exists a term of U equal to 3 ru , (c) all the terms of the geometric progression 23,,,aa a … are terms of U . If a and d are not required to be positive integers, show, by giving a counter-example, that the result in part (i) is not necessarily true. 11. [DHS-EJC/2019/Q3(part)] (a) Let x , y, z be real numbers such that 0. xyz<<< Show that 222, , xyz are three consecutive terms of an arithmetic sequence if and only if 111, , y zx zx y+++ are three consecutive terms of another arithmetic sequence. [3] (b) An infinite sequence of integers, , ,nan +∈  is defined by 12 1aa= = and 2 1 1 2, 2.n n n aan a + − += ≥ (i) Show that 2 11 1 n nn n nn a aa a aa + +− + ++ = for 2.n≥ [2] (ii) Hence or otherwise, show that 11 4n nna aa+− = − and explain why na is an odd integer for all .n +∈  [2] 12. [2018/Q1] For any positive integer n, the function Fn is defined by ( ) ( ) ( ) ( )( ) ( ) 22 1 22 41 1 F() 1 11 1 n r n r rx r r x x r r r x rx= − + +− + = + −+ +∑ . (i) Find F (0)n and describe the behaviour of F (0)n as n tends to infinity. [5] (ii) It is given