Question

Do the following sequences converge or diverge? If it converges, find its limit.

a) a_{n =} (4n^3+3n-6) / (5n^26n+2)

b) a_{n} = (3n^3+2n-6) / (4n^3+n^2+3n+1)

c) a_{n} = (n sin n) / (n^2+4)

d) a_{n} = (1/5)^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

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)

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

Write out the first 6 terms in each case and establish where
they converge or diverge
asubn= sqrt(n)sin(pi/n), n=1,2,3,4,5,6
bsubn= sin((n/n+3)^2n), n=1,2,3,4,5,6
sqrt(5),sqrt(5sqrt(5)),sqrt(5sqrt(5sqrt(5)))

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

Find the point-wise limit and find out if the sequence converges
uniformly on [0,1] to its point-wise limit
1) (X^n)/(n)
2) (X^n)/(1-X^n)

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?

Find the sum of the following series:∞∑n=2 5^n/13^(n+3)
Determine whether the following series are con-vergent or not.
Show your work!∞∑n=2 5n^2+ 5n/7n^2+ 3n+ 6

Find the pointwise limit of the following sequences of
functions on the segment [0,2]. Is this convergence uniform on
[0,2] a) fn(x)=(x^n)/((x^n)+1

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