Chapter 10 Part 1

Question 1

if a(n) does not have a finite limit, we say the sequence is _______

Answer 1

divergent

Question 2

if a(n) is a sequence of real numbers, then an infinite series is an expression of the form:

Answer 2

∞
∑ a(k) = a(1) + a(2) + … + a(n) + …
k=1

Question 3

a(n)

Answer 3

the nth or general term of the series

Question 4

p-series

Answer 4

∞
∑ 1 / k^p
k=1

Question 5

a p-series with p = 1 is called _____ series

Answer 5

harmonic

Question 6

p-series converges if

Answer 6

p>1

Question 7

p-series diverges if

Answer 7

p is less than or equal to 1

Question 8

geometric series

Answer 8

has a first term (a) and a common ration of terms (r)
∞
∑ ar^k-1
k=1
or
∞
∑ ar^k
k=0

Question 9

geometric series converges only if

Answer 9

|r| < 1

Question 10

geometric series will converge to S

Answer 10

S = a / 1-r

Question 11

if ∑ a(k) converges, then
lim (n->∞) a(n) =

Answer 11

0

Question 12

a finite number of terms MAY NOT be added to or deleted from a series without ALSO affecting its convergence or divergence (T/F)

Answer 12

false
- the terms of a series may be multiplies by a nonzero constant

Question 13

if ∑a(n) and ∑b(n) both converge, so does

Answer 13

∑[a(n) + b(n)]

Question 14

if lim (n->∞) a(n) does not equal zero, then ∑a(n) ______

Answer 14

diverges

Question 15

non-negative series

Answer 15

where a(n) is greater than or equal to 0 for all n

Question 16

if f(x) is a continuous, positive, decreasing function and f(n) = a, then ∑a(n) converges only if

Answer 16

∞
∫ f(x) dx converges
1

Question 17

if ∑a(n) converges and a(n) is greater than or equal to u(n), then ∑u(n) ______

Answer 17

converges (goes to a finite number, like 0)

Question 18

if ∑u(n) diverges and a(n) is greater than or equal to u(n), the ∑a(n) ______

Answer 18

diverges (goes to an infinite number, like infinity)

Question 19

Limit Comparison Test (not working with non-negative series)

Answer 19

if lim (n->∞) a(n)/b(n) = L, where 0<L<∞ (positive finite number), then ∑a(n) and ∑b(n) both converge or diverge

Question 20

if lim (n->∞) a(n)/b(n) = 0, and ∑b(n) converges, then ∑a(n) _______

Answer 20

converges

Question 21

if lim (n->∞) a(n)/b(n) = ∞, and ∑b(n) diverges, then ∑a(n) _______

Answer 21

diverges

Question 22

Ratio Test (∑a(n) as a non-negative series)

Answer 22

• let lim (n->∞) |a(n+1)/a(n)| = L, if it exists
• ∑a(n) converges if L<1 and diverges if L>1
• if L = 1, this test is inconclusive

Question 23

Root Test

Answer 23

• let lim (n->∞) (a(n))^1/n = L, if it exists
• ∑a(n) converges if L<1 and diverges if L>1
• if L = 1, this test is inconclusive

Question 24

telescoping series

Answer 24

• sum of series is whatever doesn’t cancel out
• lim (n->∞) b(n+1) = 0

Question 25

n-th term test

Answer 25

infinite series ∑a(n) diverges if lim (n->∞) a(n) does not = 0

Question 26

integral test

Answer 26

- a(n) = f(x) - f(x) must be positive, decreasing, and continuous - check if decreasing first - then take ∫ (1->∞) f(x)dx - either both converge or diverge

Question 27

limit comparison test

Answer 27

- find b(n) to compare - take limit (n->∞) |a(n)/b(n)| = L - L must be finite and positive to use test - both converge or diverge

Question 27

direct comparison test

Answer 27

- choose b(n) to compare to a(n) - test for convergence/divergence of b(n) then check if b(n) is larger or smaller than a(n) - if b(n)>a(n), b(n) must be convergent to use test - if a(n) is larger, must be divergent to use test - both converge or diverge

Question 28

alternating series test

Answer 28

- a(n) must be positive - converges if: 1. lim (n->∞) a(n) = 0 2. a(n+1) less than or equal to a(n)