RUA C8

Question 1

Geometric Series(Σ c * r^(n-1) )

Answer 1

|r| < 1 -> converges S = a(1) ( 1/(1-r) )
|r| >= 1 -> diverges

Question 2

Theorem: If Σ a(n) is convergent…

Answer 2

then lim as n-> infity of a(n) = 0

Question 3

Test for Divergence: If lim as n->infity of Σ a(n)…

Answer 3

≠ 0 or DNE -> divergent
else inconclusive

Question 4

Theorem: If Σ a(n) and Σ b(n) are convergent…

Answer 4

1) Σ ( a(n) ± b(n) ) = Σ a(n) + Σ b(n)
2) Σ c * a(n) = c Σ a(n), where c is a constant

Question 5

P-Series( Σ 1/(n^p) )

Answer 5

p > 1 -> converges
p <= 1 -> diverges

Question 6

Telescoping Series(Σ a(n))…if lim as n->infity of S(n)…

Answer 6

= L ->converges
= ± infity or DNE -> diverges

Question 7

Integral Test
a(n) = f(n)
where f(n) must be positive, continuous, and decreasing [N,infity]
so
∫ (from 1-infity) f(x) dx……

Answer 7

= L -> converges
= ± infity or DNE -> diverges

Question 8

Ratio Test
lim as n -> infity of |a(n+1))/a(n)|…

Answer 8

< 1 -> converges
>1 or infity -> diverges
= 1 -> inconclusive

Question 9

Root Test
lim as n -> infity of (a(n) )^(1/n)…

Answer 9

< 1 -> converges
>1 ->diverges
= 1 -> inconclusive

Question 10

Direct Comparison
If Σ a(n) & Σ b(n) are series w/ positive terms

Answer 10

1) if Σ b(n) is convergent & >= a(n) for all n, then Σ a(n) is convergent
2) if Σ b(n) is divergent & <= a(n) for all n, then Σ a(n) is divergent

Question 11

Limit Comparison Test
If Σ a(n) & Σ b(n) are series w/ positive terms
If lim as n -> infit of (a(n)/b(n))….

Answer 11

= C
where c is finite number and >0, then both series converge or diverge

Question 12

Alternating Series Test
If alternating series b(n) satisfies….

Answer 12

1) b(n+1) <= b(n) for all n
2) lim as n -> infity of b(n) = 0
then the series is conditionally convergent