test 3

Question 1

Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears
to converge, make a conjecture about its limit. If the sequence diverges, explain why:
an+1 = 1n/10, a0=1

Answer 1

1/10, 1/100, 1/1000, 1/10000
0; converges

Question 2

Consider the formulas for the following sequences {an}∞
n=1. Determine a plausible limit
of the sequence or state that the sequence diverges:
an=2^n sin(2^-n)

Answer 2

1; converges

Question 3

Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears
to converge, make a conjecture about its limit. If the sequence diverges, explain why:
an+1 = 1 – an/2, a0=2/3

Answer 3

2/3; converges

Question 4

When a biologist begins a study, a colony of prairie dogs has a population of 250.
Regular measurements reveal that each month the prairie dog population increases by
3%. Let pn be the population (rounded to whole numbers) at the end of the nth-month,
where the initial population is p0 = 250.

Question 5

Consider the formulas for the following sequences {an}∞
n=1. Determine a plausible limit
of the sequence or state that the sequence diverges
an= (100n^2 -1) / (10n)

Answer 5

diverges

Question 6

Consider the following infinite series. 1 / 2^k
(3.1) Find the first four partial sums S1, S2, S3, and S4 of the series.
(3.2) Find a formula for the nth-partial sum Sn of the infinite series. Use this formula
to find the next four partial sums: S5, S6, S7, and S8 of the infinite series.
(3.2) Make a conjecture for the value of the series
sigma 1 / 2^k

Question 7

Consider the following infinite series.
(3.1) Find the first four partial sums S1, S2, S3, and S4 of the series.
(3.2) Find a formula for the nth-partial sum Sn of the infinite series. Use this formula
to find the next four partial sums: S5, S6, S7, and S8 of the infinite series.
(3.2) Make a conjecture for the value of the series
sigma 2 / 3^(k-1)

Question 8

Find the limit of the following sequences or determine that the sequence diverges
{cot( (npi) / (2n+3)}

Question 9

formula for nth

Answer 9

(a (1-r^n)) / 1-r

Question 10

how to divide fractions ?

Answer 10

multiply by the reciprocal

Question 11

how to get r and a for nth formula ?

Answer 11

r: 2nd divided by 1st term
a= first term

Question 11

Find the limit of the following sequences or determine that the sequence diverges
NOT YET

Question 11

Find the limit of the following sequences or determine that the sequence diverges
{ (3^n) / (3^n + 4^n)}

Question 11

Find the limit of the following sequences or determine that the sequence diverges
{5(1.01)^n}

Question 12

Find the limit of the following sequences or determine that the sequence diverges
{(1/n)^1/n}

Question 12

Sleep model After many nights of observation, you notice that if you oversleep one night, you tend to undersleep the following night, and vice versa. This pattern of
compensation is described by the relationship
xn+1 = 1/2 (xn + xn−1), for n = 1, 2, 3, …,
where xn is the number of hours of sleep you get on the nth-night, and x0 = 7 and
x1 = 6 are the number of hours of sleep on the first two nights, respectively.

Write out the first six terms of the sequence xn and confirm that the terms alternately
increase and decrease

Show that the explicit formula xn = 19/3 + 2/3 (− 1/2)^n
for n ≥ 0 generates the terms
of the sequence in part (a)

C. assume the limit of the sequence exists. What is the limit of the sequence?

Question 13

Express each sequence {an}∞
n=1 as an equivalent sequence of the form {bn}∞ n=3
{2n+1}∞ n=1

Question 14

Express each sequence {an}∞
n=1 as an equivalent sequence of the form {bn}∞ n=3
{n^2+6n-9}∞ n=1

Question 15

Evaluate each geometric series or state that it diverges
sigma: (3 times 4^k) / (7^k)

Question 16

Evaluate each geometric series or state that it diverges
sigma: 1/16 + 3/64 + 9/256 + 27/1024….

Question 17

Evaluate each geometric series or state that it diverges
sigma:
3e^-k

Question 18

Evaluate each geometric series or state that it diverges
sigma: 1 / (9k^2 + 15k + 4)

Question 19

Evaluate the series
∞∑
k=0
( 4 / 3^k) − (4/3^(k+1))
Use a telescoping series argument.
(b) Use a geometric series argument.

Question 20

Evaluate the series
∞∑
k=0
( 4 / 3^k) − (4/3^(k+1))
Use a geometric series argument.

Question 21

Use the Divergence Test to determine whether the following series diverge or state that
the test is inconclusive
∞∑
k=1
(k^3) / (k^3 + 1)

Question 22

Use the Divergence Test to determine whether the following series diverge or state that
the test is inconclusive
∞∑
k=1
(k^2) / (2^k)

Question 23

Use the Divergence Test to determine whether the following series diverge or state that
the test is inconclusive
∞∑
k=1
k^(1/k)

Question 24

Integral test:
∞∑
k=1
1 / (2k+4)^2

Question 25

Integral test:
∞∑
k=1
(e^k) / (1+e^(2k))

Question 26

Integral test:
∞∑
k=1
1 / k(lnk)^2

Question 27

Integral or divergence test or p-series:
∞∑
k=1
1 / √k^3

Question 28

Whats the p-series ?

Question 29

Whats the integral test?

Question 30

whats the divergence test?

Question 31

Integral or divergence test or p-series:
∞∑
k=1
(√k^(2) +1) / k

Question 32

Integral or divergence test or p-series:
∞∑
k=1
1 / k(lnk)lnlnk

Question 33

Integral or divergence test or p-series:
∞∑
k=1
1 / 3√27k^2

Question 34

Choose your test:
∞∑
k=1
ln ( (2k^6) / (1+k^6) )

Question 35

Use the Comparison Test or the Limit Comparison Test to determine whether the
following series converge.
(a)
∞∑
k= 1
(k^2 + k − 1) /
(k^4 + 4k^2 − 3)

Question 36

Use the Comparison Test or the Limit Comparison Test to determine whether the
following series converge.
(a)
∞∑
k= 1
1 / (5^k +3) NO

Question 37

Use the Comparison Test or the Limit Comparison Test to determine whether the
following series converge.
(a)
∞∑
k= 1
(k^2 + k + 2) / ( 6^k (k^2 + 1 )

Question 38

Use the Comparison Test or the Limit Comparison Test to determine whether the
following series converge.
(a)
∞∑
k= 1
√ ( (k) / (k^3 + 1) )

Question 39

Use the Comparison Test or the Limit Comparison Test to determine whether the
following series converge.
(a)
∞∑
k= 1
1 / 3^k - 2^k

Question 40

Use the Comparison Test or the Limit Comparison Test to determine whether the
following series converge.
(a)
∞∑
k= 1
1 / ( k √(k+2)

Question 41

choose:
∞∑
k= 1
(tan-1k) / (k^2)

Question 42

high negative + low positive ?

Answer 42

substract and keep negative

Question 43

high negative - low positive ?

Answer 43

add and keep negative

Question 44

low negative - high negative ?

Answer 44

subtract, positive

Question 45

low negative + high negative ?

Answer 45

add, negative