UNIT 10] series convergence/divergence

Question 1

what does a geometric series look like?

Answer 1

∑ ∞ n=0 (a)(r^n)

Question 2

when is a geometric series convergent?

Answer 2

when |r|<1

Question 3

when is a geometric series divergent?

Answer 3

when |r|>=1

Question 4

how do you find the sum of a geometric series if it converges?

Answer 4

(a)/(1-r)

Question 5

what does a series that the nth-term test works on look like?

Answer 5

∑ ∞ n=0 (a_n)

Question 6

can an nth-term test be used to find convergence?

Answer 6

no

Question 7

when is a nth-term series divergent?

Answer 7

when lim n→∞ (a_n) =/= 0, or when it does not exist.

Question 8

what happens in an nth term series if the limit is 0?

Answer 8

there is no conclusion

Question 9

what does a series that the integral test can be used on look like?

Answer 9

∑ ∞ n=N (a_n)

Question 10

when is an integral test series convergent?

Answer 10

∫[N, ∞] (f(x))dx = L

Question 11

when is an integral test series divergent?

Answer 11

∫[N, ∞] (f(x))dx = ∞

Question 12

what does the graph of f(x) have to behave like for an integral test?

Answer 12

f(x) has to be positive, decreasing, and continuous

Question 13

what is the behavior of the series like when doing an integral test?

Answer 13

the behavior of the series and integral are the same

Question 14

what does a p-series look like?

Answer 14

∑ ∞ n=N (1)/(n^p)

Question 15

when is a p-series convergent?

Answer 15

when p>1

Question 16

when is a p-series divergent?

Answer 16

when 0<p<=1

Question 17

what is a p-series called when p=1?

Answer 17

a harmonic series

Question 18

does the harmonic series always diverge or converge?

Answer 18

diverge

Question 19

if a series is an airplane, what is divergence?

Answer 19

when the plane lands

Question 20

if a series is an airplane, what is convergence?

Answer 20

when the plane is gaining altitude

Question 21

what does a series that you can use the direct comparison test on look like?

Answer 21

∑ (a_n)

Question 22

when is a direct comparison test series convergent?

Answer 22

when a_n <= c_n

Question 23

when is a direct comparison test series divergent?

Answer 23

when a_n>= c_n

Question 24

what are the two functions being compared in a direct comparison test?

Answer 24

∑ (a_n) (original) and ∑ (c_n)

Question 25

in a direct/limit comparison test, what do the nth term expressions need to be

Answer 25

> 0, positive

Question 26

in a direct/limit comparison test, what do you need to know about the new series?

Answer 26

if it converges or diverges

Question 27

when is a limit comparison series convergent

Answer 27

when lim n→∞ ((a_n)/(c_n)) = 0

Question 28

when is a limit comparison test series divergent?

Answer 28

when lim n→∞ ((a_n)/(c_n)) = ∞