Series

Question 1

Arithmetic Sequence

Answer 1

General form:
f(n) = a + (n-1)d

Recursive form:
f(n+1) = f(n) + d

a = first term
d = common difference
n = input

Question 2

Geometric Sequence

Answer 2

General form:
f(n) = ar^n-1

Recursive form:
f(n+1) = f(n)*r

a = first term
r = common ratio
n = input

Question 3

Harmonic Sequence

Answer 3

General form:
f(n) = 1/[a + (n-1)d]

Recursive form:
f(n+1) = f(n) + d

Question 4

Arithmetic Series (Sn)

Answer 4

n/2 [2a + (n-1)d]

Question 5

Geometric Series (Sn)

Answer 5

a(1-r^n)
————
1 - r

Question 6

What is Test 1 for convergence?

Answer 6

If lim n->∞ Un = 0, the series may converge or diverge

If lim n->∞ Un ≠ 0, the series diverges

Question 7

What is test 2 for convergence

Answer 7

1 1 1 1
— + — + — + —
1^p 2^p 3^p 4^p

p > 1 = converges
p ≤ 1 = diverges

Question 8

Test 3 for convergence

Answer 8

Lim Un + 1
n->∞ ——— =
. Un

< 1 = converges
> 1 = diverges
= 1 = dk

Question 9

What is the difference between absolute and conditional convergence

Answer 9

absolutely convergent = If Σ|Un| converges

conditionally convergent = If Σ|Un| doesn’t converge, but ΣUn

Question 10

Maclaurin’s Series

Answer 10

f(x) ≡ f(0) + xf’(0) + x^2 f’’(0)/2! + x^3 f’’’(0)/3! + …

Question 11

Taylor’s Series

Answer 11

f(x+a) ≡ f(a) + xf’(a) + x^2 f’’(a)/2! + x^3 f’’’(a)/3! + …

Question 12

How do you calculate an indeterminate using L’Hôpital’s rule

Answer 12

Lim f(x) Lim f’(x)
x->a —— = x->a ——
. g(x) g’(x)

• carry on differentiating until you get a non 0 or ∞ value