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Homework Statement
This is the question of mine that I'm having a little confusion about. I know the whole process in which you use the ratio test to determine the radius of convergence and using that you test the end points of the summation to see if they converge at the end points aswell...
Homework Statement
Show that if \vartheta is any constant not equal to 0 or a multiple of 2\pi, and if u_{0}, u_{1}, u_{2} is a series that converges monotonically to 0, then the series \sum u_{n} cos(n\vartheta +a) is also convergent, where a is an arbitrary constant.
Homework...
Homework Statement
Determine if the series the summation form n=2 to infinity of n/((n2+1)ln(n2+1)) is convergent or divergent.
Homework Equations
The Attempt at a Solution
I applied the integral test and got positive infinity, so I it diverges. But I want to know if I'm right...
Homework Statement
Find the radius and interval of convergence for the power series of n=0 of infinity of n^3(x-5)^n
Homework Equations
Ratio test: http://en.wikipedia.org/wiki/Ratio_test
The Attempt at a Solution
[(n+1)^3(x-5)^n+1 / n^3(x-5)^n]
I am lost as to how to...
Homework Statement
Does the series:
sum from 1 to inf. (n+2)/(n+1)
converge? If so does it converge absolutely?
Homework Equations
Ratio test for series
The Attempt at a Solution
I found this series to converge using the root test, (.jpg attached) however wolfram alpha...
Homework Statement
Does ∫dx/sqrt(x^4+1) from x=-∞ to x=∞ converge or diverge?
explain in detail if you can please.
thanks
Homework Equations
limit comparison test
direct comparison
The Attempt at a Solution...well i have the answer, it converges. I just need a better...
Homework Statement
Hi, So i don't know if this is a stupid question but i'll ask anyways. So I'm on the chapter where we start testing integrals for convergence. The books starts out with elementary functions then they move towards non elementary functions. Testing for them is OK, my problem...
EDIT: On pg. 390 of Kreyszig's Functional Analysis text, we have: "If T is a bounded linear operator on a nonempty Banach space, then the series
\sum_{k=0}^\infty \left( \frac{T}{\lambda} \right)^k
converges absolutely for |\lambda| > 2\| T \|."
The argument presented in Kreyszig...
Homework Statement
For which positive integers k is the following series convergent? (To enter - or , type -INFINITY or INFINITY.)
Summation of n=1 to infinity of (n!)^2 / (kn)!
Homework Equations
ratio test: limit n-->infinity of [((n+1)!)^2/(kn+1)!] / [(n!)^2 / (kn)!] (have the...
Homework Statement
I'm tasked with integrating the following functions, and values for t where the function converges:
\int_{0}^{1}x^p\cdot ln(x) dxHomework Equations
Integration by Parts Formula: \int udv=uv-\int vdu
The Attempt at a Solution
I found a definite integral...
Homework Statement
A function f is defined by...
f(x) = \frac{n+1}{3^{n+1}} x^n
a.) find the interval of convergence of the given power series.
b.) Find \lim_{x\rightarrow 0} \frac{f(x) - \frac{1}{3}}{x}
c.) Write the first three nonzero terms and the general term for...
Homework Statement
Determine the radius of convergence, the interval of convergence, and
the sum of the series
Summation from k=2 to ∞ of
k(x-2)^k+1.
Homework Equations
ratio test? The Attempt at a Solution
possibly take the derrivitive of the power series, then find the sum then integrate...
Homework Statement
I need to find the domain of absolute convergence of the following series:
^{\infty}_{1}\sum(z+3)^{2n}/(2n)!
Homework Equations
Ratio test?
The Attempt at a Solution
I'm not really sure how to handle the complex variable z within the series. I attempted to use the...
Homework Statement
http://im2.gulfup.com/2011-04-01/1301686351321.gif
Homework Equations
superlinearly convergence
The Attempt at a Solution
[PLAIN][PLAIN]http://im2.gulfup.com/2011-04-01/1301686616101.gif
this is what i know about it, kindly help me
Homework Statement
\Sigma (from index k = 1 until infinity)
Within the Sigma is the series : (k! * (x^k))
Homework Equations
Ratio Test : lim as k approaches infinity |a(k+1) / ak|
The Attempt at a Solution
When I apply the ration test to the series and simplify I get lim k...
Homework Statement
Suppose that: sum [a_n (n-1)^n] is the Talyor series representation of tanh(z) at the point z = 1. What is the largest subset of the complex plane such that this series converges?
Note: 'sum' represents the sum from n=0 to infinity
Homework Equations
tanh(z) =...
Homework Statement
Discuss the convergence of
Integral(x3/2 sin 2x dx) range is from 0 to 1Homework Equations|sin x| =< 1
The Attempt at a Solution
The sine function converges absolutely. It is also increasing from 0 to 90 degrees, decreases until 27 decrease and it is negative from 180 to...
Homework Statement
Say that
\sum_{k=1}^{\infty }a_k
converges and has positive terms. Does the following necessarily converge?
\sum_{k=1}^{\infty }{a_k}^{5/4}
Homework Equations
If it necessarily converges, a proof is required, if not, a counter-example is required.
The...
Homework Statement
Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations
Cauchy inequalities (estimates from the Cauchy integral formula)The...
Find the convergent sum and find the sum of first five terms
\sum_{n=1}^{\infty} \frac{sin(nx)}{2^nn}
from 1 to infinity.
I have found so far that:
\sum_{n=1}^{\infty} \frac{sin(nx)}{2^n} = \frac{2sin(x)}{5-4cos(x)}
I am not sure how to consider the \frac{1}{n} term.
Can someone please help?
Homework Statement
Determine if the following converge:
a. (∞,n=1) ∑ (1+1/n)^n
b. (∞,n=1) ∑ sin(n)/(n^2 + √n)
c. (∞,n=3) ∑ 1/(k ln^2 k)
The Attempt at a Solution
a. I tried the root test, but it failed, so i immediately went to the limit divergence test...ended up getting...
Homework Statement
Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
\sumn=1infinity (-3)n-1/4nHomework Equations
A geometric series, \sumn=1infinity arn-1=a + ar + ar2 + ... is convergent if |r|< 1 and its sum is \sumn=1infinity arn-1 =...
Homework Statement
Determine whether the sequence converges or diverges. If it converges, find the limit.
an = e1/n
Homework Equations
The limit laws, adapted for sequences.
The Attempt at a Solution
I have the solution; I was just wondering if someone might explain it to me.
I...
Homework Statement
Does the series ((-1)^n*n!)/(1*6*11*...*(5n+1)) from n = 0 to \infty
absolutely converge, converge conditionally or diverge?
Homework Equations
The Attempt at a Solution
I did the ratio test for ((-1)^n *n!)/(5n+1)) and I found that it diverges but apparently...
Homework Statement
\Sigma from n=0 to infinity (-10)n/n!
Determine the absolute convergence, convergence, or divergence of the series.
Homework Equations
In this section, it's suggested that we use the following to determine a solution:
A series is called absolutely convergent if the series...
Homework Statement
\sum from n=1 to inf (1+ 1/2 + ... 1/n)x^n
Find the radius of convergence and the interval of convergence of the given power series.
Homework Equations
Dunno..
The Attempt at a Solution
Stuck thinking about it. I'm not sure if I can combine what's in brackets with the...
Homework Statement
For what values of r does \int(from 0 to infinity) xre-x dx converge?
I assume that the problem refers to r as any real number.
2. The attempt at a solution
I have given this a try but I am really not confident that I did it right...
First i used integration...
F: C(Omega) -> D'(Omega); F(f) = F_f
--
O = Omega
Introduce the notion of convergence on C(Omega) by
f_p -> f as p -> inf in C(O) if f_p(x) -> f(x) for any xEO
Show that then F is a continuous map from C(O) to D'(O)
Hint: Use that if a sequence of continuous functions converges to a...
Homework Statement
If the sequence xn ->a , and the sequence yn -> b , then xn - yn -> a - b
The Attempt at a Solution
Can someone check this proof? I'm aware you cannot subtract inequalities, but I tried to get around that where I indicated with the ** in the following proof...
Homework Statement
Test for convergence
\sum sin(1/n) / \sqrt{ln(n)}
sum from 2 to inf
Homework Equations
Limit comparison test
if lim an / bn to inf
doesn't equal 1 you know if it converges or diverges
by limit comparison test
The Attempt at a Solution
I've tried a lot of different...
Homework Statement
show if has a uniform convergence of pointwise
also we know that x gets values from 0 to 1The Attempt at a Solution
for the pointwise I think its easy to show that limfn(x) as n->infinity is 0
but I am really stuck in uniform convergence
I know that fn converges...
My functional analysis professor made the following assertion the other day: If f_n \to f in the L^2 norm, then there is a subsequence f_{n_k} that converges pointwise almost everywhere to f. This is the first I've heard of that...can someone point me to a proof of this proposition? Does it have...
Homework Statement
Homework Equations
[PLAIN]http://img577.imageshack.us/img577/6756/physforumsquestion.png
The Attempt at a Solution
I am pretty much stuck guys. Any help will be greatly appreciated!
Homework Statement
Let r be a positive number and define F = {u in R^n | ||u|| <= r}. Use the Componentwise Convergence Criterion to prove F is closed.Homework Equations
The Componentwise Convergence Criterion states: If {uk} in F converges to c, then pi(uk) converges to pi(c). That is, the...
Hello, I am preparing for a screening exam and I'm trying to figure out some old problems that I have been given.
Given:
Suppose f is contained in L1([a,b])
Prove for almost everywhere x is contained in [a,b]
limit as h goes to 0+, int (abs(f(x+t)+f(x-t)-2f(x)))dt = 0
Initially I...
Homework Statement
Approximate the sum of the series S = \sum(n from 1 to Infinity) \frac{[(-1)^(n+1)]}{n!} by calculating S_10.
Estimate the level of error involved in this problem.
AND
S = \sum(n from 1 to Infinity) \frac{[(-1)^(n+1)]}{n^4}
Approximate the sum of the series by...
Homework Statement
Show that there are continuous functions g:[-1,1]\to R such that no sequence of polynomials Q_n satisfies Q_n(x^2)\to g(x) uniformly on [-1,1] as n\to\infty
The Attempt at a Solution
Suppose there is a sequence Q_n such that Q_n(x^2)\to g(x) uniformly for g(x)=x.
Then...
Homework Statement
[PLAIN]http://img153.imageshack.us/img153/4822/radiusm.jpg
Homework Equations
The Attempt at a Solution
Using the ratio test:
\left | \frac{e^{i(n+1)^2 \theta} \theta^{n+1} z^{(n+1)^2}}{e^{in^2 \theta} \theta ^n z^{n^2}} \right |
= | \theta...
Homework Statement
[PLAIN]http://img225.imageshack.us/img225/7501/complexh.jpg
Homework Equations
The Attempt at a Solution
How do I show this absolutely converges?
Find the value of x so that the series below converge.[/b]
\sum 1/ [(k^x) * (2^k)] (k=1 to \infty)
Using ratio test, I 've got
[(1/2) * 1^x] < 1 for all x in R
But when I use different value of x, series converge and diverge!
Really need your help! Thank you so much
I know that the sequences meets the following:
(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)
I've got the feeling that this sequence is alternating or decreasing, but I was unable to prove it.
Usually I use induction to prove things about such inductive sequence but in this case I don't have real values t o...
Hello,
I have question about using Dirichlet's Convergence Test which states:
1. if f(x) is monotonic decreasing and \lim_{x\rightarrow \infty} f(x)=0
2. G(x)=\int_a^x g(t)dt is bounded.
Then \int_a^\infty f(x)g(x)dx is convergent.
But what about the following situation:
f(x)=1/x
g(x)=cosxsinx...
Homework Statement
Hello, I'm trying to revise for my probability exam next week and am getting a bit hung up on how to show a sequence of random variables converges.
For example, how would I got about doing this question:
Let {X_{n}:n\geq1} be a sequence of random variables with...
Homework Statement
Ok, so I don't need help with this part, I just got stuck at the following step when attempting to find the interval of convergence:
The Attempt at a Solution
I got here:
-4 < x^2 < 4
So, I need to solve this inequality. But can I? How can I take the square root...
Hi. Can I have some help in answering the following questions? Thank you.
Let {f_n} be a sequence of functions from N(set of natural numbers) to R(real nos.) where
f_n (s)=1/n if 1<=s<=n
f_n (s)=0 if s>n.
Define f:N to R by f(s)=0 for every s>=1...
It's been a while since I've done rate of convergence problems,
how would i find the rate of convergence for either of these sequences?
1) limn->infsin(1/n)=0
2)limn->infsin(1/n^2)=0
Homework Statement
let an be a positive series.
it is known that for every bn\rightarrow \infty
the sum from 1 to inf of an/bn is convergent
prove that the sum from 1 to inf of an is convergent
Homework Equations
The Attempt at a Solution
I thoght maybe to try to say. let...
it's a new semester, and we're at it again. series abounds!
Homework Statement
Let .
Determine whether the following series converges:
Homework Equations
-1 \leq \sigmak \leq 1
The Attempt at a Solution
i feel like the series inherently diverges because of the 1/k element...
Homework Statement
Find the radius of convergence of the power series:
a) \sum z^{n!}
n=0 to infinity
b) \sum (n+2^{n})z^{n}
n=0 to infinity
Homework Equations
Radius = 1/(limsup n=>infinity |cn|^1/n)
The Attempt at a Solution
a) Is cn in this case just 1? And plugging it in...
Hi. I have been banging my head against this problem for a while and I just don't get it. Maybe (probably) it's something wrong with my logarithm-fu or limit-fu. I just registered to ask this because I couldn't find an answer anywhere, and I've been reading these forums for a while for other...