Ch. 10 Sequences and Series - Precalculus

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Convergent sequence

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Tags and Description

23 Terms

1

Convergent sequence

Has a limit such that the terms approach a unique number

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2

Divergent sequence

The terms don’t approach a finite number

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3

Series

  • Indicated sum of all the terms of a sequence

    • Finite & infinite

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4

nth Partial Sum (Sₙ)

The sum of the first n terms of a series

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5

Recursive formula for arithmetic sequence

aₙ = aₙ₋₁ + d

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6

Explicit formula for arithmetic sequence

aₙ = a₁ + (n-1)d

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7

Arithmetic sequence with 2nd differences

  • Quadratic

  • aₙ = an² + bn + c

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8

nth Partial Sum of an Arithmetic Series

  • Sₙ = (ⁿ/₂)(a₁ + aₙ)

  • Sₙ = (ⁿ/₂)[2a₁ + (n-1)d]

<ul><li><p>Sₙ = (ⁿ/₂)(a₁ + aₙ)</p></li><li><p>Sₙ = (ⁿ/₂)[2a₁ + (n-1)d]</p></li></ul>
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9

Recursive formula for geometric sequence

aₙ = aₙ₋₁・r

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10

Explicit formula for geometric sequence

aₙ = a₁・rⁿ⁻¹

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11

nth Partial Sum of a Geometric Series

  • Sₙ = a₁[(1-rⁿ)/(1-r)]

  • Sₙ = (a₁-aₙr)/(1-r)

<ul><li><p>Sₙ = a₁[(1-rⁿ)/(1-r)]</p></li><li><p>Sₙ = (a₁-aₙr)/(1-r)</p></li></ul>
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12

Sum of an Infinite Geometric Series

S = a₁/(1-r)

<p>S = a₁/(1-r)</p>
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13

Pascal’s Triangle

A triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y)ⁿ

  • Recursive: coefficients in the (n-1)th row can be added together to find coefficients in the nth row

<p>A triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression, such as (x + y)ⁿ</p><ul><li><p>Recursive: coefficients in the (n-1)th row can be added together to find coefficients in the nth row</p></li></ul>
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14

Formula for Binomial Coefficients of (a + b)ⁿ

ₙCᵣ = n!/[(n-r)!r!]

  • For the aⁿ⁻ᣴbᣴ term

<p>ₙCᵣ = n!/[(n-r)!r!]</p><ul><li><p>For the aⁿ⁻ᣴbᣴ term</p></li></ul>
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15

Formula for Binomial Experiments of (p + q)ⁿ

ₙCₓ・pˣ qⁿ⁻ˣ

  • x = successes

  • n = # of trials

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16

Binomial Theorem

(a + b)ⁿ = ₙC₀ aⁿb⁰ + ₙC₁ aⁿ⁻¹b¹ + ₙC₂ aⁿ⁻²b² + … + ₙCᵣaⁿ⁻ᣴbᣴ + ₙCₙ a⁰bⁿ

  • r = 0, 1, 2, … , n

<p>(a + b)ⁿ = ₙC₀ aⁿb⁰ + ₙC₁ aⁿ⁻¹b¹ + ₙC₂ aⁿ⁻²b² + … +  ₙCᵣaⁿ⁻ᣴbᣴ + ₙCₙ a⁰bⁿ</p><ul><li><p>r = 0, 1, 2, … , n</p></li></ul>
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17

Power Series

  • Infinite

  • x & aₙ take on any values for n = 0, 1, 2, …

<ul><li><p>Infinite</p></li><li><p>x &amp; aₙ take on any values for n = 0, 1, 2, …</p></li></ul>
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18

Exponential Series

  • Infinite

  • Represents eˣ

<ul><li><p>Infinite</p></li><li><p>Represents eˣ</p></li></ul>
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19

Power Series for cos x

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20

Power Series for sin x

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21

Euler’s Formula

eⁱᶿ = cos θ + i sin θ

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22

Exponential Form of a Complex Number

a + bi = r × eⁱᶿ

  • r = √(a² + b²)

  • θ = tan⁻¹(b/a); a > 0

  • θ = tan⁻¹(b/a) + π; a < 0

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23

Natural Logarithm of a Negative Number

iπ = ln (-1)

  • ln (-k) = ln [(k)(-1)]

    • ln (k) + ln (-1)

    • ln k + iπ

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