[H2 MATH] Chapter 4 - Series _ Summation

SHALYN TAY COPYRIGHTED © A LEVEL H2 MATHEMATICS SEQUENCES & SERIES (APGP & SUMMATION)

CHAPTER ANALYSIS MASTERY EXAM WEIGHTAGE SHALYN TAY COPYRIGHTED ©

Sequences & Series SHALYN TAY COPYRIGHTED © A sequence is a set of numbers arranged in a defined order. Each number in the sequence is called a term of the sequence, 𝑢𝑛 𝑢1,𝑢2,𝑢3 , … , 𝑢𝑛 Sequence The entire set of ordered elements is a sequence Term of the sequence is usually denoted by 𝒖𝒏 Types of Sequences: • Finite: sequence terminates • Infinite: there is no end, sequence keeps going on… • Convergent: 𝑢𝑛 approaches a unique value (the limit) as 𝑛→∞ • Divergent: sequence that does not converge • Strictly increasing: 𝑢𝑛+1>𝑢𝑛 for all 𝑛 • Strictly decreasing: 𝑢𝑛+1<𝑢𝑛 for all 𝑛 • Constant: 𝑢𝑛=𝑐 for all 𝑛 A series is the sum of the terms of a sequence, denoted by 𝑆𝑛 . It can have a finite or infinite number of terms. Useful results: Series 𝑆𝑛=𝑢1+𝑢2+𝑢3 + … + 𝑢𝑛 Sum of a Sequence 𝑢𝑛=𝑆𝑛−𝑆𝑛−1 for 𝑛>1 𝑢1=𝑆1

ARITHMETIC PROGRESSION GEOMETRIC PROGRESSION SEQUENCES & SERIES PART I SHALYN TAY COPYRIGHTED ©

Arithmetic Progression (AP) SHALYN TAY COPYRIGHTED © An arithmetic progression is a sequence of numbers for which the difference between every 2 consecutive terms is a constant, known as the common difference, d: 𝑢𝑛−𝑢𝑛−1=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑓𝑜𝑟𝑎𝑙𝑙𝑣𝑎𝑙𝑢𝑒𝑠𝑜𝑓𝑛>1 Proof for Arithmetic Progressions 𝑢𝑛−𝑢𝑛−1=𝑑 The nth term of the arithmetic progression is: Sum of first n terms of an arithmetic series is: 𝑆𝑛= 𝑛 2 2𝑎+ 𝑛−1 𝑑 = 𝑛 2 (𝑢1+𝑢𝑛) 𝑢𝑛=𝑎+ 𝑛−1 𝑑 First Term Common DifferenceTerm of AP (common difference, d) To prove a sequence is an AP , show: “Since the difference between 2 consecutive terms is a constant, the terms of this sequence form an arithmetic progression.” “Since 𝑢3−𝑢2≠𝑢2−𝑢1 the terms of the sequence do not form an arithmetic progression.”

Geometric Progression (GP) SHALYN TAY COPYRIGHTED © An geometric progression is a sequence of numbers for which the ratio of every 2 consecutive terms is a constant, known as the common ratio, r: 𝑢𝑛 𝑢𝑛−1 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑓𝑜𝑟𝑎𝑙𝑙𝑣𝑎𝑙𝑢𝑒𝑠𝑜𝑓𝑛>1 Proof for Geometric Progressions 𝑢𝑛 𝑢𝑛−1 =𝑟 The nth term of the geometric progression is: Sum of first n terms of a geometric series is: 𝑆𝑛= 𝑎(𝑟𝑛−1) 𝑟−1 = 𝑎(1−𝑟𝑛) 1−𝑟 , 𝑟≠1 𝑢𝑛=𝑎𝑟𝑛−1 First Term Common RatioTerm of GP(common ratio, r) To prove a sequence is an GP , show: “Since the ratio of every 2 consecutive terms is a constant, the terms of this sequence form a geometric progression.” “Since 𝑢3 𝑢2 ≠ 𝑢2 𝑢1 the terms of the sequence do not form a geometric progression.” Sum to infinity of a geometric series: 𝑆∞= 𝑎 1−𝑟 , 𝑟 <1

SUMMATION OF SERIES METHOD OF DIFFERENCE SEQUENCES & SERIES PART I SHALYN TAY COPYRIGHTED ©

Summation of Series SHALYN TAY COPYRIGHTED © The Sigma Notation: 𝑟=1 𝑛 𝑎=𝑛𝑎 Standard Results 𝑟=𝑘 𝑛 𝑢𝑟=𝑢𝑘+𝑢𝑘+1+⋯+𝑢𝑛−1+𝑢𝑛 r is a dummy variable known as the index of summation k is the lower limit of the summation or the starting value of r n is the upper limit of the summation or the ending value of r When the serie