Question

Determine whether the following sequences converge or diverge. If it converges, ﬁnd the limit. Must show work

1.)an = nsin(1/n)

2.)an = sin(n)

3).an =4^n /1 + 9^n

4).an = ln(n+1) − ln(n)

Answer #1

Determine whether the following sequences converge or diverge.
If a sequence converges, find its limit. If a sequence diverges,
explain why.
(a) an = ((-1)nn)/
(n+sqrt(n))
(b) an = (sin(3n))/(1- sqrt(n))

Use L’Hopital’s Rule to determine if the following sequences
converge or diverge. If the sequence converges, what does it
converge to?
(a) an = (n^2+3n+5)/(n^2+e^n)
(b) bn = (sin(n −1 ))/( n−1)
(c) cn = ln(n)/ √n

Do the following sequences converge or diverge? If it converges,
find its limit.
a) an = (4n^3+3n-6) / (5n^26n+2)
b) an = (3n^3+2n-6) / (4n^3+n^2+3n+1)
c) an = (n sin n) / (n^2+4)
d) an = (1/5)^n

prove whether the following series converge absolutely, converge
conditionally or diverge give limit
a) sum of (-5)n /n! from 0 to infinity
b) sum of 1/nn from 0 to infinity
c) sum of (-1)n /(1 + 1/n) from 0 to infinity
d) sum of 1/5n from 0 to infinity

Determine whether the sequence converges or diverges. If it
converges, find the limit. (If an answer does not exist, enter
DNE.)
an = (4^n+1) /
9^n

Determine whether the limit converges or diverges, if it
converges, find the limit.
an = (1+(4/n))^n

Determine whether the sequence converges or diverges. If it
converges, find the limit. (If an answer does not exist, enter
DNE.) a n = n 3 /n + 2

Determine whether the sequence converges or diverges. If it
converges, find the limit. (If an answer does not exist, enter
DNE.)
an = 4 − (0.7)n
lim n→∞ an =
please box answer

Explain whether the following integrals converge or not. If the
integral converges, find the value. If the integral does not
converge, describe why (does it go to +infinity, -infinity,
oscillate, ?)
i) Integral from x=1 to x=infinity of x^-1.4 dx
ii) Integral from x=1 to x=infinity of 1/x^2 * (sin x)^2 dx
iii) Integral from x=0 to x=1 of 1/(1-x) dx

Use the ratio test to determine whether∑n=12∞n2+55n
converges or diverges.
(a) Find the ratio of successive terms. Write your
answer as a fully simplified fraction. For n≥12,
limn→∞∣∣∣an+1an∣∣∣=limn→∞
(b) Evaluate the limit in the previous part. Enter ∞
as infinity and −∞ as -infinity. If the limit does
not exist, enter DNE.
limn→∞∣∣∣an+1an∣∣∣ =
(c) By the ratio test, does the series converge,
diverge, or is the test inconclusive?

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