[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 series is infinite, n is replaced with ∞ A series has 𝒏−𝒌+𝟏 terms 𝑟=𝑘 𝑛 𝑎𝑢𝑟±𝑏𝑣𝑟=𝑎 𝑟=𝑘 𝑛 𝑢𝑟±𝑏 𝑟=𝑘 𝑛 𝑣𝑟 𝑟=𝑘 𝑛 𝑢𝑟 = 𝑟=1 𝑛 𝑢𝑟− 𝑟=𝑘 𝑘−1 𝑢𝑟 Properties of Sigma Notation: for 2≤𝑘≤𝑛 𝑟=1 5 1 3𝑟=1 3+1 9+ 1 27+ 1 81+ 1 243E.g. You may also use the G.C. to find the sum of a series directly [math > summation] where a and b are constants 𝑟=1 𝑛 𝑟=1 2 𝑛(𝑛+1) 𝑟=1 𝑛 𝑟2=1 6 𝑛 𝑛+1 (2𝑛+1) 𝑟=1 𝑛 𝑟3=1 4 𝑛2(𝑛+1)2= 1 2 𝑛(𝑛+1) 2 = 𝑟=1 𝑛 𝑟 2 Arithmetic Progression 𝑟=1 𝑛 [𝑎+ 𝑟−1 𝑑]=𝑛 2 2𝑎+ 𝑛−1 𝑑 Geometric Progression 𝑘=1 𝑛 𝑎𝑟𝑘−1=𝑎(𝑟𝑛−1) 𝑟−1

Method of Difference SHALYN TAY COPYRIGHTED © The method of difference is used to evaluate sums where the term 𝑢𝑟 can be expressed as a difference of two or more terms which will result in the cancellation of most terms leaving only a final solution. Convergence / Divergence of Series 𝑟=1 𝑛 𝑢𝑟= 𝑟=1 𝑛 (𝑓 𝑟 −𝑓 𝑟−1 ) = (𝑓 1 −𝑓 0 ) + 𝑓 2 −𝑓 1 + (𝑓 3 −𝑓 2 ) ⋮ + (𝑓 𝑛−2 −𝑓 𝑛−3 ) + (𝑓 𝑛−1 −𝑓 𝑛−2 ) + (𝑓 𝑛 −𝑓 𝑛−1 ) =(𝑓 𝑛 −𝑓 0 ) Many questions usually require the use of partial fractions to express the term ur into the difference of more terms. Typically, we can also denote 𝑟=1 𝑛 𝑢𝑟 by 𝑆𝑛 . If 𝑆𝑛 approaches a unique value as n approaches infinity, we say 𝑆𝑛 converges and the sum to infinity 𝑆∞ exists. Or else it is said to diverge. 𝑆∞= 𝑎 1−𝑟 , 𝑟 <1 Geometric Series: 𝑟=1 𝑛 1 𝑟− 1 𝑟+1 = ( 1 1− 1 2) + 1 2− 1 3 + ( 1 3− 1 4) ⋮ + ( 1 𝑛−2− 1 𝑛−1) + ( 1 𝑛−1− 1 𝑛) + ( 1 𝑛− 1 𝑛−1) =( 1 1− 1 𝑛−1) As 𝑛 approaches ∞ , 1 𝑛−1 →0 , 𝑟=1 ∞ 1 𝑟− 1 𝑟+1 →1−0=1 Sum to Infinity:

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