Question

Summation from n=2 to infinity of ln(n)/n^2 * x^n

a.) Let x=-1, and compute the integral test to determine whether this it is convergent or divergent

b.) Compute the ratio test to determine the interval of convergence and explain what this interval represents.

I'm really confused with this problem (specifically a)- please write out all of the details of computing the integral so I can understand. Thank you!

Answer #1

A) Use the Comparison Test to determine whether integral from 2
to infinity x/ sqrt(x^3 -1)dx is convergent or divergent.
B)Use the Comparison Test to determine whether the integral from
2 to infinity (x^2+x+2)/(x^4+x^2-1) dx is convergent or
divergent.

I'm trying to solve for sequence and series question. The
sequence goes from n=2 to infinity and an =
1/n(ln(n))4/3. I've been trying to figure out using
limit form of comparison test. However, when I used 1/n or 1/ln(n)
as ab I get convergence and 1/n and 1/ln(n) are both
divergent by proof.

6. Let series {an} = 1/(n2 + 1) and series {bn} = 1/n2.
Use Limit Comparison Test to determine if each series is convergent
or divergent.
7. Use Ratio Test to determine if series {an}= (n +
2)/(2n + 7) where n is in interval [0, ∞]
is convergent or divergent. Note: if the test is
inconclusive, use n-th Term Test to answer the
question.
8. Use Root Test to determine if series {an} = nn/3(1 +
2n) where n...

Use the integral test to determine the divergence or convegence
of the series (1/ (ln(5))^n) ) I know it to be Convegence, unsure
how its convergent.

1) find the Taylor series expansion of
f(x)=ln(x) center at 2 first then find its associated radius of
convergence.
2) Find the radius of convergence and interval
of convergence of the series Σ (x^n)/(2n-1) upper infinity lower
n=1

Find a general term (as a function of the variable n) for the
sequence{?1,?2,?3,?4,…}={45,1625,64125,256625,…}{a1,a2,a3,a4,…}={45,1625,64125,256625,…}.
Find a general term (as a function of the variable n) for the
sequence {?1,?2,?3,?4,…}={4/5,16/25,64/125,256/625,…}
an=
Determine whether the sequence is divergent or convergent. If
it is convergent, evaluate its limit.
(If it diverges to infinity, state your answer as inf . If it
diverges to negative infinity, state your answer as -inf . If it
diverges without being infinity or negative infinity, state your
answer...

Sigma(1, infinity) (2^n x^n)/(n!) : Radius of convergence of
this series?

find the Interval of convergence of
1. sum from {0}to{infinity} X ^ 2n / 3^n

1. Given that 1 /1−x = ∞∑n=0 x^n with convergence in (−1, 1),
find the power series for x/1−2x^3 with center 0.
∞∑n=0=
Identify its interval of convergence. The series is convergent
from
x=
to x=
2. Use the root test to find the radius of convergence for
∞∑n=1 (n−1/9n+4)^n xn

Given the alternating series:
sigma(2 to infinity): (-1)^n / ln n
Determine if the series converge
absolutely. (Use the fact
that: ln n <
n)
Determine if the series converge
conditionally.
(Estimate the sum of the infinite series using the
first 4 terms in the series and estimate the
error.
How many terms should we use to approximate the sum of
the infinite series in question, if we want the error to be less
than 0.5?

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