What is Comparison test: Definition and 105 Discussions
In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.
Attempt:
Note we must have that
## f>0 ## and ## g>0 ## from some place
or
## f<0 ## and ## g<0 ## from some place
or
## g ,f ## have the same sign in ## [ 1, +\infty) ##.
Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...
The answer sheet states that the series converges by limit comparison test (the second way).
In the case of this particular problem, would it be also okay to use the comparison test, as shown above? (The first way)
Thank you!
Use the comparison test to determine if the series series convergences or divergences
$$S_{6}=\sum_{n=1}^{\infty} \dfrac{1}{n^2 \ln{n} -10}$$
ok if i follow the example given the next step alegedly would be...
$$\dfrac{1}{n^2 \ln{n} -10}<\dfrac{1}{n^2 \ln{n}}$$
$\tiny{242 UHM}$
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with the proof of Theorem 2.3.9 (a) ...
Theorem 2.3.9 reads as follows:
Now, we can prove Theorem 2.3.9 (a) using the Cauchy...
Homework Statement
Homework EquationsThe Attempt at a Solution
So the book is saying that this series diverges, i have learned my lesson and have stopped doubting the authors of this book but i don't understand how this series diverges. ok i can use the comparison test using 1/3n and 1/3n...
The problem
In this problem I am supposed to show that the following series converges by somehow comparing it to ## \frac{1}{k\sqrt{k}} ## :
$$ \sum^{\infty}_{k=1} \left( \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} \right) $$
The attempt
## \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} =...
Homework Statement
##\sum _{n=0}^{\infty }\:\sin \left(\frac{1}{n}\right)##
Homework Equations
The Attempt at a Solution
Can I try comparison test by
##\left(\frac{1}{1+n}\right)<sin\left(\frac{1}{n}\right)##
since ##\left(\frac{1}{1+n}\right)## diverges also...
Homework Statement
Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is:
The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞
The question is also attached as a .png file
2. Homework Equations
The...
Homework Statement
I know that ∑n=1 to infinity (sin(p/n)) diverges due using comparison test with pi/n, despite it approaching 0 as n approaches infinity.
However, an alternating series with (-1)^n*sin(pi/n) converges. Which does not make sense because it consists of two diverging functions...
Homework Statement
Hi
I am looking at the proof attached for the theorem attached that:
If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2##
where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##.
For any integer ##r \geq 0 ## :
##\Omega_r := {mw_1+nw_2|m,n \in...
Homework Statement
Use a comparison test to determine whether this series converges:
\sum_{x=1}^{\infty }\sin ^2(\frac{1}{x}) Homework EquationsThe Attempt at a Solution
At small values of x:
\sin x\approx x
a_{x}=\sin \frac{1}{x}
b_{x}=\frac{1}{x}
\lim...
Homework Statement
\sum_{x=2}^{\infty } \frac{1}{(lnx)^9}
Homework EquationsThe Attempt at a Solution
x \geqslant 2
0 \leqslant lnx < x
0 < \frac{1}{x} < \frac{1}{lnx}
From this we know that 1 / lnx diverges and I wanted to use this fact to show that 1 / [(lnx) ^ 9] diverges but at k...
Homework Statement
Determine whether the series converges or diverges.
∞
∑ 1/n!
n=1
Homework Equations
If ∑bn is convergent and an≤bn for all n, then ∑an is also convergent.
Suppose that ∑an and ∑bn are series with positive terms. If
lim an = C
n→∞ bn
where c is finite number and c>o...
The limit comparison test states that if $a_n$ and $b_n$ are both positive and $L = \lim_{{n}\to{\infty} } \frac{a_n}{b_n} > 0$ then $\sum_{}^{} a_n$ will converge if $\sum_{}^{} b_n$ and $\sum_{}^{} a_n$ will diverge if $\sum_{}^{} b_n$ diverges. Does this rule also apply if $L$ diverges to...
I have this series:
$$\sum_{k = 1}^{\infty} {4}^{\frac{1}{k}}$$
To solve this, I am trying to compare it to this series
$$\sum_{k = 1}^{\infty} {4}^{k}$$
So, I can let $a_k = {4}^{\frac{1}{k}} $ and $b_k = {4}^{k}$
These seem to be both positive series and $ 0 \le a_k \le b_k$
Therefore...
Hi everybody! I have another question about integrability, especially about the limit comparison test. The script my teacher wrote states:
(roughly translated from German)
Limit test: Let -∞ < a < b ≤ ∞ and the functions f: [a,b) → [0,∞) and f: [a,b) → (0,∞) be proper integrable for any c ∈...
Hey,
I am working on Calculus III and Analysis, I really need help with this one problem. I am not even sure where to begin with this problem. I have attached my assignment to this thread and the problem I need help with is A. Thank you!
Use the comparison test to see if \sum_{1}^{\infty}{\left[n\left(n+1\right)\right]}^{-\frac{1}{2}} converges?
I tried n+1 \gt n, \therefore n(n+1) \gt n^2 , \therefore {\left[n(n+1)\right]}^{\frac{1}{2}} \gt n, \therefore {\left[n(n+1)\right]}^{-\frac{1}{2}} \lt \frac{1}{n} - no conclusion...
Homework Statement
use the comparison theorem to determine whether ∫ 0→1 (e^-x/√x) dx converges.
Homework Equations
I used ∫ 0 → 1 (1/√x) dx to compare with the integral above
The Attempt at a Solution
i found that ∫ 0 → 1 (1/√x) dx = 2 ( by substituting 0 for t and take the limit of the...
I have the sum,
$$\sum_{n=1}^{\infty} \frac{1}{n^{3}}\sin(n \pi x) \text{, where }0 \leq x \leq 1$$
I have to show that the series converges, so I'm going to use the Comparison Test.
$$ \text{If }0 \leq a_n \leq b_n \text{ then}$$$$\text{If }\sum b_n \text{ converges then }\sum a_n \text{ must...
Homework Statement
Determine whether the series is converging or diverging
Homework Equations
∞
∑ 1 / (3n +cos2(n))
n=1The Attempt at a Solution
I used The Comparison Test, I'm just not sure I'm right. Here's what I've got:
The dominant term in the denominator is is 3n and
cos2(n)...
In my textbook it says if you are comparing limn->infinity of an/bn an>0 and bn>0 for the limit comparison test to apply.
It says nothing about "an" having to be greater than "bn", so as long as both are positive for each term I can use the limit comparison test right? It isn't like the...
Homework Statement
Use the limit comparison test to prove convergence or divergence for the series sum from n=1 to infinity for ((5n^3)+1)/((2^n)((n^3)+n+1))
Homework Equations
The limit comparison test says that if you have two positive series, sum An and sum Bn, let C=lim n to infinity of...
Homework Statement
Use the limit comparison test to show the series converges or diverges: Sum from n=1 to infinity of ((5n^3)+1)/((2^n)((n^3)+n+1))
Homework Equations
suppose Sum An and Sum Bn are two positive series. Let lim as n goes to infinity of An/Bn = c: 1) if 0<c<inifinity then either...
The question asks whether the following converges or diverges.
\int_{0}^{\infty } \frac{\left | sinx \right |}{x^2} dx
Now I think there might be a trick with the domain of sine function but I couldn't make up my mind on this.
I tried to compare it with 1/x^2, (sinx)/x, and sinx. I actually...
Homework Statement
Use the limit comparison test to check for convergence or divergence: Sum from n=1 to infinity of ((2n)^2+5)^-3
Homework Equations
let lim n to infinity of An/Bn = c
1) if 0<c<infinity then either both converge or both diverge
2) if c=0 and sum Bn converges, so does sum An...
\int_{0}^{\infty} \frac{x^2 dx}{x^5+1}
The question asks whether this function diverges or converges.
I have tried to do some comparisons with x^2/(x^6+1), and x^2/(x^3+1) but it didn't end up with something good.
Then I decided to compare it with \frac{x^2}{x^4+1}
Since this function...
K≥0 ∑ ((sqrt(k)+2)/(k+5))
I am trying to prove that this diverges. The divergence test is inconclusive.
Now I am left with a great option of a comparison test. I'm not quite sure what to compare it with, but I know I need to compare it with something smaller (denominator is larger) that...
I'm trying to find if this series converges or diverges using the comparison test:
and the answer goes:
My problem is, I am not sure how to go from 1/2^(n+1) to 1/2(1/2)^n.
can you please explain that to me
Homework Statement
Use the comparison test to show that the series converges, and find the value it converge to by using partial fractions.
∑ n=1 -> ∞: \frac{2}{n^2 + 5n + 6}
Homework Equations
The Attempt at a Solution
The series can be written as 2 * ∑ n=1 -> ∞...
Homework Statement
Use $$\sum\limits_{n=1}^∞ \frac{1}{n^2}$$ to prove by the comparison test that $$\sum\limits_{n=1}^∞ \frac{n+1}{n^3} $$ converges.Homework Equations
$$\sum\limits_{n=1}^∞ \frac{n+1}{n^3} \equiv \sum\limits_{n=1}^∞ \frac{1}{n^2} + \sum\limits_{n=1}^∞ \frac{1}{n^3} $$
The...
Homework Statement
$$\sum\limits_{n=1}^∞ \frac{1}{n√(n)} $$
Since $$ \frac{1}{n√(n)} \equiv \frac{1}{x^{3/2}} $$ this is a convergent p-series. But, when I attempt to prove this by the limit comparison test with known convergent series such as $$\sum\limits_{n=1}^∞ \frac{1}{n^2}$$
ex...
Hey, not too sure about what function i would compare this integral from 1 to infinity of (3x^3 -2)/(x^6 +2) dx. I also have to show that it converges.
Thanks!
Homework Statement
Does \sum_{n=1}^{\infty}a_n where a_n = \frac{(n+1)^{1/3}-n^{1/3}}{n} converge or diverge?
Homework Equations
The Attempt at a Solution
The ratio test is inconclusive, as is the root test. The limit is equal to 0, but that doesn't say much. I've tried to find...
Homework Statement
Use the Limit Comparison Test to determine if the series converges or diverges:
Ʃ (4/(7+4n(ln^2(n))) from n=1 to ∞.
(The denominator, for clarity, in words is: seven plus 4n times the natural log squared of n.)
Homework Equations
Limit Comparison Test:
Let Σa(n) be the...
If we have two sequences and the ratio of their limit is greater than zero, why does this mean that they either both converge or diverge? I don't understand why the test works.
Also, what about lim[(1/x)/(1/x^2)] = lim x = ∞?
The series of 1/x^2 converges but series of 1/x diverges...
Homework Statement
Determine whether Ʃ(n from 1 to infinity) ln(n)/n^3 converges or diverges using the limit comparison test.
Homework Equations
I must use the limit comparison test to solve this problem-not allowed to use other tests.
The Attempt at a Solution
I know that the...
1. Homework Statement [/b]
Use the direct comparison test to show that the following are convergent:
(a)\int_1^∞ \frac{cos x\,dx}{x^2}
I don't know how to choose a smaller function that converges similar to the one above. The main problem is i don't know where to start.
A simple...
would someone please care to reword this proof for me?
http://en.wikipedia.org/wiki/Limit_comparison_test
it talks about ε, which is not even defined and then n0, which is again not defined, what the hell are all these variables... I'm sure someone here could do a better job organizing...
Use the comparison test to find out whether or not the following improper integral exist(converge)?
integral(upper bound:infinity lower bound:2) 1/(1-x^2) dx
Here's my solution for 3),but I think something went wrong
For all x>=2
0<=-(2-2x)<=-(1-x^2) that means: 0<=-1/(1-x^2)<=-1/(2-2x)...
Learning about the Limit Comparison Test for Improper Integrals. I haven't gotten to any applications or actual problems yet. Just learning the theory so far, and have a question on the very beginning of it.Homework Statement
f(x) ~ g(x) as x→a, then \frac{f(x)}{g(x)} = 1
(that is, f(x)...
Homework Statement
Show that:
\sum \frac{3}{n^{2} + 1}
converges from n = 1 to ∞
Homework Equations
If Ʃbn converges, and Ʃan < Ʃbn.
Ʃan also converges.
The Attempt at a Solution
\sum \frac{1}{n^{2}} converges
\sum \frac{3}{n^{2} + 1} = 3 * \sum \frac{1}{n^{2} + 1}...
Homework Statement
I need to use the Comparison Test or the Limit Comparison Test to determine whether or not this series converges:
∑ sin(1/n^2) from 1 to ∞
Homework Equations
Limit Comparison Test: Let {An} and {Bn} be positive sequences. Assume the following limit exists:
L =...
Homework Statement
Is the series convergent or divergent? \sum_{n=0}^{\infty}{\frac{1}{\sqrt{n+1}}}
Homework Equations
I can use any test but wolfram alpha says that it is divergent by comparison test.
The Attempt at a Solution
How do I apply comparison test?
I can compare it to: \sum _{...
Homework Statement
Use any test to determine whether the series converges.Homework Equations
\displaystyle \sum^{∞}_{n=1} tan(1/n) The Attempt at a Solution
Direct Comparison Test
tan(1/n) > 1/n
By integral test: 1/n diverges thus, by dct, tan(1/n) diverges.
Homework Statement
use limit comparison test.
Homework Equations
\displaystyle\int_2^∞ {\frac{1}{\sqrt{x^2 - 1}} dx}
The Attempt at a Solution
I have tried usin 1/x as the comparison function, but when applying the test it
comes out to 0, not an L -> 0 < L < ∞
Homework Statement
Use direct comparison test or limit comparison test to determine if the integral converges.Homework Equations
\displaystyle\int_0^6 {\frac{dx}{9-x^2}}
The Attempt at a Solution
If i were to use the limit comparison test, would these integrals fit the criteria.
** if the...
Homework Statement
A few examples done in class that I didn't understand are as follows:
(Find out if convergent or divergent)
integral 0-->infinity ( x / (x^3 + 1) )
integral 1-->infinity ( (2+e^(-x) ) / x )
Homework Equations
None
The Attempt at a Solution
Basically, I have NO idea...