Sequences and Series

Exercise Find the first four terms of the sequence 3 1, 1, 2, 3, .

[Answer: 2, 5, 8, 11]

Example Let be the sequence defined by 2, 1, and

21 13 2 for 0,1,2,, find and . Solution:

(k = 0) 21 13 2 2 2 3

(k = 1) 32 24 2 6 2 8 ⁄ Exercise Without using a calculator, find the first 4 terms of the recursively defined sequence

• 5, 2 3 [Answer: 5, 7, 11, 19] • 4, 1 [Answer: 4, , , ]

• 2, 3, [Answer: 2, 3, 5, 8] Example Evaluate ∑ 0.5 and round your answer to 5 decimal places. ! Solution:

∑ 0.5 !

0.5 0.5 0.5 ! ! !

0.5 0.5 ! ! 0.5 0.5 0.5 0.5 0.5 0.5 0.041666 0.003125 0.000186 0.000009 . Exercise Find and evaluate each of the following sums • ∑ [Answer: 225] • ∑ 1 5 [Answer: 521]

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Sequences and Series

Arithmetic Sequences

an = an‐1 + d, also an = a1 + (n – 1)d n S = ()a +a n 2 1 n Exercise • For the arithmetic sequence 4, 7, 10, 13, (a) find the 2011th term; [Answer: 6,034] (b) which term is 295? [Answer: 98th term] (c) find the sum of the first 750 terms. [Answer: 845,625] rd th • The 3 term of an arithmetic sequence is 8 and the 16 term is 47. Find and d and

construct the sequence. [Answer: 2,3;the sequence is 2,5,8,11,] • Find the sum of the first 800 natural numbers. [Answer: 320,400] • Find the sum ∑ 4 5. [Answer: 34,710]

Geometric Sequences

n‐1 an = ran‐1, also an = a1 × r a (1− r n ) S = 1 n 1− r a Infinite Geometric Series: For r < 1, S = 1 1− r Example Find the exact value of the geometric series . Solution: (Check that the given series is a geometric series by evaluating ratio of consecutive terms) ⁄ 5 ; ⁄ ⁄ 5 ; ⁄ geometric series with common ratio (Since || 1, apply the infinite sum formula for geometric series) ⁄ ⁄ ⁄ ⁄

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Sequences and Series

Exercise • Find the exact value of the geometric series 2 1 . [Answer: ] • Find the 8th term of the geometric sequence 4, 20, 100, . [Answer: 312,500] • Find the 10th term of the geometric sequence 64, 32, 16, 8, . [Answer: ] • Find the sum of the first 10 terms of the geometric sequence 3, 15, 75, 375, . [Answer: 7,324,218] • Find the sum ∑ 0.4 and round your answer to 6 decimal places. [Answer: 0.666597] • Find the value of the infinite series ∑ . [Answer: ]

Maclaurin Series f ''()0 x 2 f '''(0)x3 f (n)(0)x n • f ()x = f ()0 + f '(0)x + + +"+ +" 2! 3! n!

• 1 all ! ! • sin all ! ! • cos 1 all ! ! • ln1 for || 1

n nn( −1) • (Binomial Series) ()11+=++xnxx2 +" () x < 1 2! Example Find the first three non-zero terms of the Maclaurin series for . √ Solution: (Method 1: Use the definition – straight forward but tedious, in comparison with method 2) 1⁄ 0 10⁄ 1⁄ 1 first non zero term; 1⁄ · 2 1⁄ 0 10⁄ 0 this zero term doesnt count;

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Sequences and Series

1⁄ · 1⁄ · 2

⁄ 1⁄ 1⁄ 21 5 ⁄ ⁄ 2 2 5 3 2 3 21⁄ 0 3·0 210⁄ 21 second non zero term 61⁄ 3 2 · 1⁄ · 2 61⁄ 3 21⁄ 1⁄ 41 33 2 1⁄ 44 9 6 1⁄ 10 5 1⁄ 2 10⁄ 20 0 this zero term doesnt count; 2 1⁄ 231⁄ 2 · 1⁄ · 2 231⁄ 2 1⁄ · 0 23·010⁄ 2·0010⁄ third non zero term

Hence 1 √ ! ! 1 (Method 2: Use the binomial series and replace x and n by and respectively)

⁄ 1 1 1 √ 1

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Sequences and Series

Example Find the power series of tan centered at zero. Solution: (Since tan , we can find the power series of tan by integrating the power series of , which can be obtained by using the Binomial series) 1 ∑ ! · ∑ ∑ 1 ! tan ∑ 1 ∑ 1 ∑ 1 (To determine C, evaluate both sides at a convenient value of x, say x = 0)

tan 0∑ 1 000,hence ∑ Exercise • By direct expansion, find the first four nonzero terms of the Maclaurin expansion for 1. [Answer: 1 443 ] • Find the first three non-zero terms of the Maclaurin series of sin3. [Answer: sin3 3 ] • Find the first four non-zero terms of the Maclaurin series of cos by multiplying the Maclaurin series for and cos . [Answer: cos 1 ] • Estimate ln 0.96 by using four terms of the expansion for ln1. [Answer: 0.0408220] • Find the first three nonzero terms of the expansion for by using the binomial √ series. [Answer: 1 ] √

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Sequences and Series

Taylor Series

! ! ! Example Approximate √9.6 by using the second Taylor polynomial of √ centered at 9. Solution: √ √9 3; ⁄ ; √ ⁄ ; √

! ⁄ √9.6 9.6 9.6 3 9.6 9 9.6 9 ! 3 0.6 0.36 . Exercise • Find the first three nonzero terms of the Taylor expansion for cos, with .

[Answer: cos √ ] • Expand in a Taylor series centered at 1. [Answer: 1 1 1 ] • Expand √ in powers of 4. [Answer: √ 2 4 4 4 ]

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