Sequences and Series

Monotonic Sequences

Strictly increasing or strictly decreasing sequence:

• Strictly Increasing: a_n≤a_n+1 or a_n/a_n+1≤1

• Strictly Decreasing: a_n≥a_n+1 or a_n/a_n+1≥1

Or take the first derivative and check sign

Bounded Sequence

• Bounded Above: a_n≤M

• Bounded Below: a_n≥M

• Bounded: N≤a_n≤M

Geometric Series:

• Infinite

• Finite

Form: arⁿ

If |r| > 1 series converges:

• Infinite: Series converges to ar^k/1-r ar^k = first term

• Finite: Series converges to: a(1-rⁿ)/1-r n = distance between bounds

Divergence Test

lim a_n=0 TFD is inconclusive otherwise series diverges

Integral Test

Conditions: f(n) must be positive, continuous, and decreasing

If the integral from 1 to infinity of f(x) = a finite number then the series converges

lim b→infinity of the intergal from lower bound to b

P-Series

Converges when p>1

Diverges when p≤1

Direct Comparison Test

0≤a_n≤b_n

1. If b_n converges, a_n also converges

2. If a_n diverges, b_n also diverges

Limit Comparison Test

If both series are positive if limit a_n/b_n = a finite number then both series must either converge or diverge

Alternating Series Test

Forms (-1)ⁿa_n or (-1)ⁿ⁺¹a_n provided a_n>0

Conditions:

1. Passes Divergence Test: lim = 0

2. Decreasing Sequence

If these are true the series is convergent

Absolute and Conditional Convergence

1. If |a_n| is convergent and a_n is also convergent, a_n is absolutely convergent

2. If |a_n| diverges but a_n converges, a_n is conditionally convergent

Ratio Test

Good for factorials

1. Absolutely convergent: lim |a_n+1/a_n|= L < 1

2. Divergent: lim |a_n+1/a_n|= L > 1

3. Inconclusive: lim |a_n+1/a_n|= L = 1

Root Test

Good for series raised to the n

Good for factorials

1. Absolutely convergent: lim |a_n+1/a_n|= L < 1

2. Divergent: lim |a_n+1/a_n|= L > 1

3. Inconclusive: lim |a_n+1/a_n|= L = 1

Taylor

f(x) = f(c )+f’(c )(x-c)+ f’’(c )(x-c)²/2! …

How to calculate radius of covenrgence

Use 1/R = Ratio Tes