Ch. 10 Sequences and Series - Precalculus

Convergent sequence

Convergent sequence

Has a limit such that the terms approach a unique number

Divergent sequence

The terms don’t approach a finite number

Series

• Indicated sum of all the terms of a sequence

  • Finite & infinite

nth Partial Sum (Sₙ)

The sum of the first n terms of a series

Recursive formula for arithmetic sequence

aₙ = aₙ₋₁ + d

Explicit formula for arithmetic sequence

aₙ = a₁ + (n-1)d

Arithmetic sequence with 2nd differences

• Quadratic

• aₙ = an² + bn + c

nth Partial Sum of an Arithmetic Series

• Sₙ = (ⁿ/₂)(a₁ + aₙ)

• Sₙ = (ⁿ/₂)[2a₁ + (n-1)d]

Recursive formula for geometric sequence

aₙ = aₙ₋₁・r

Explicit formula for geometric sequence

aₙ = a₁・rⁿ⁻¹

nth Partial Sum of a Geometric Series

• Sₙ = a₁[(1-rⁿ)/(1-r)]

• Sₙ = (a₁-aₙr)/(1-r)

Sum of an Infinite Geometric Series

S = a₁/(1-r)

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

Formula for Binomial Coefficients of (a + b)ⁿ

ₙCᵣ = n!/[(n-r)!r!]

• For the aⁿ⁻ᣴbᣴ term

Formula for Binomial Experiments of (p + q)ⁿ

ₙCₓ・pˣ qⁿ⁻ˣ

• x = successes

• n = # of trials

Binomial Theorem

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

• r = 0, 1, 2, … , n

Power Series

• Infinite

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

Exponential Series

• Infinite

• Represents eˣ

Power Series for cos x

Power Series for sin x

Euler’s Formula

eⁱᶿ = cos θ + i sin θ

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

Natural Logarithm of a Negative Number

iπ = ln (-1)

• ln (-k) = ln [(k)(-1)]

  • ln (k) + ln (-1)

  • ln k + iπ