Sequence Definition and 1000 Threads

Homework Statement Find the limit of the sequence \lim n\rightarrow \infty \frac{e^n+3^n}{5^n} Homework Equations The Attempt at a Solution L'Hopital's rule never ends with this one. But even after taking the first derivative of the top and bottom it shows that 5x will always be grower...

I know about the proof using lim inf and lim sup and the proof using a convergent subsequent, however I thought about this proof. Can you tell me if it is correct, and if not why? Thank you let Sn be Cauchy seq in R Let S be its range. Then S is bounded. Since R is complete, sup...

Homework Statement Find the general term of the sequence, starting with n=1, determine whether the sequence converges, and if so find its limit. (a) {(1/2), (3/4), (5/6), (7/8), ...} (b) {(1-(1/2), ((1/3)-(1/2)), ((1/3)-(1/4)), ((1/5)-(1/4)),...} Homework Equations My text only...

Homework Statement Hello! I'm having some trouble trying to understand basic concepts of Convex Analysis (I study it independently). In particular, I have a book (Convex Analysis and Optimization - Bertsekas) which gives a definition for the convergence of a sequence of sets: Homework...

(23)/(24), (11)/(12), (21)/(24),(5)/(6), (19)/(24), (3)/(4)

I don't speak English very well, so it's very hard to me to explain my attemps to solve this problem, and I'm still learning to use latex, so it's so slow to me. I can scan my attemps if you want to see them. Homework Statement I_n = \int_{0}^{\infty} x^{2n-1}/(x^2+1)^{n+3} \dx, n \geq 1 I...

Can someone explain to me what is this sequence referred in the page? http://oeis.org/A004730"

Homework Statement Can someone explain to me what is this sequence referred in the page? http://oeis.org/A004730"

Homework Statement Prove that if the sequence {An} converges to A, then the sequence {|An|} converges to |A|. And is the converse true? This is for my calculus class and it needs to be in proof format. Thank you! The Attempt at a Solution I'm totally lost, I was going to use ||x| -...

Homework Statement Determine the limit of the sequence (n^n)/(n!) Homework Equations The Attempt at a Solution I think the limit should be infinity as n^n grows faster than n!, but I'm not sure how to prove it. Thanks for the help!

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...

If some (sequence)^2 converges does that mean the original (sequence) always converges? (using mobile version) attempted solution: All i know is if {an} converges to L==> {an}^2 converges to L^2

Dear friends can you show me please an example of a sequence of integrable functions fn:R->R converging to an integrable function f but *not* in the L1-norm, i.e. such that \Int \mid f_n -f\mid is not equal to 0? Thank u a lot

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...

Homework Statement Assume the sequence \{ a_n \} satisfys 0 < a_1 < 1 and a_{n + 1} = a_n (1 - a_n )(n \ge 1) ,prove \mathop {\lim }\limits_{n \to \infty } {\kern 1pt} {\kern 1pt} {\kern 1pt} na_n = 1 Homework Equations The Attempt at a...

Ok so for a sequence x(n) to be bounded it means |x(n)|<=M but according to by book, if x(n) belongs to some closed interval, say [a,b], x(n) is bounded. That is confusing because say x(n) belonging to [a,b] means a<=x(n)<=b. How can there exist a M such that -M<=x(n)<=M? this means...

Given f1+f3+f5+...+f2n-1 = f2n. Prove by induction. So using the general Fibonacci sequence formula, Fn=Fn-1 + Fn-2 , I got f1+f3+...+f2n+1 = f2n+2 then using the formula, f1+f3+...+f2n+1 +f2n = f2n+2. This ends up giving me, f2n + f2n does not equal f2n+2. What's wrong here? (...

I'm having trouble with a word problem: The people of Gossipopolis cannot keep a secret. Upon being told a secret, a person from Gossipopolis will spend the next hour telling three people. In turn, those friends will spend the next hour each telling 3 more people. This process continues and...

I'm a little stuck here... I need to write this in the summation notation, and then find and prove a formula in terms of n, using induction :3+7+11+...+(4n-1) I know that the summation notation is n +--- \ / 4i-1 +--- i=1 but I have no idea how to...

Homework Statement Let an = ( 1 + \frac{1}{n} )n Homework Equations show that if 0 <= a < b \frac{b n+1 - a n+1}{b-1} < (n+1)bn The Attempt at a Solution I have started from a < b and I said so an < bn Then I multiply by (n+1) So I get the left hand side...

Homework Statement Find sequence (a_n) s.t. \lim_{N\rightarrow \infty} \sum_{n=1}^{2N} a_n and \lim_{N\rightarrow \infty} \sum_{n=1}^{2N+1} a_n both converge but \sum_{n=1}^{\infty} a_n diverges. I have no idea where to start to be honest. I'm confused at how this is possible. Isn't it always...

hi all, i asked this question on y! answers and one guy gave me an advice to come here, sign up, and ask. i hope I'm in the right place! i'm a freshman majoring in mathematical econ and minor in statistics to prepare myself for the actuarial exams (and open up other career options). for this...

A finite geometric sequence has t1 = 0.1024 and t2 = 0.256. How many terms does this sequence have if its middle term has a value of 156.25? My Solution Common Ratio: T2/T1=(.256)/(.1024)=2.5 What term # is the middle term? tn=ar^n-1 a=0.1024 r=2.5 tn=156.25...

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...

I found this interesting exercise on a topology book I'm reading, but I don't have a clue what to do. Show that there is no sequence {g_n} of continuous functions from R to R such that the sequence {(g_n)(x)} is bounded iff x is rational (where R = set of real numbers).

Instead of using a sieve to remove non-primes from the sequence. 6x-1 x =0 to x=n 6x+1 x=0 to x= n What if you calculate and remove the non-primes. I have determined how to calculate the non-primes in this set. By subtracting them from the entire set you are left with all primes. I find this...

Homework Statement let (X,d) be a metric space and let A be a dense subset of X such that every Cauchy sequence in A converges in X. Prove that (X,d) is complete. Homework Equations (X,d) is complete if all Cauchy sequences in X converge. A is a dense subset of X => closure(A) = X...

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 I have to find all the partial limits {I hope this is how this term named in English} of a sequencesHomework Equations a_1=0 a_{2n}=\frac {a_{2n-1}} {3} a_{2n+1} = 1/3 + a_{2n} The Attempt at a Solution I have tried to prove first that sequences of all the even terms...

Homework Statement Show whether the series 1/sqrt(n^2 + n) converges or diverges by using the Comparison Test.Homework Equations The Attempt at a Solution It's clearly less than 1/n (divergent) which doesn't help and it's greater than 1/n^2 (convergent) [at least for large n] which doesn't...

help me i don't know where i must to posting this task if 8+4=12 7+2=9 2+3=6 6+5=10 9+5 = ?

Homework Statement I need a example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces. Homework Equations If a function f is continuous in metric space (X, d), then...

Homework Statement The sequence an = 0, if n contains the digit 9 an = 1/n. if n does not contain the digit 9 does the series\sum an converge? Homework Equations The Attempt at a Solution I have this idea to separate this series into two subseries - the harmonic and the...

if you have a sequence of events {A_1, A_2, ...} then an expression for the event that "infinitely many A_i's occur" is: U(n = 1 to inf, U(m = n to inf, A_m) ) but wouldn't U(n = 1 to inf, A_n) also satisfy this?

x_n = (n^2 / (n^2+1) , 1/sqrt(n)). prove that this sequence converges and find the limit. so as n approaches infinite it is clear that x_n approaches (1,0). so using the definition of a convergent sequence, i pick an epsilon > 0 and i have to find some N such that when n>N, sqrt( 1/(n^2 +...

Homework Statement Suppose that the following condition holds lim(n→∞)⁡ (∑ Pi ) /(n-1) = 1/2 where 0< Pi <1. Then, by l'Hopital's rule, I think the following should also hold. Right? lim(n→∞)⁡ ∑ Pi = (1/2)n + constant In this case, I am wondering whether we can...

Homework Statement Find \displaystyle\lim_{x\to\infty}x\sin(\frac{1}{x}) where x is a natural number, and this is a sequence, not a real function.Homework Equations The Attempt at a Solution I know the answer is 1, and that supposedly one solution is to introduce a dummy variable y = 1/x...

Homework Statement So the Fibonacci Sequence is defined by an = an-1 + an-2 a1=1, a2=1 We are more interested in the sequence of ratios of subsequent terms of the Fibonacci sequence define rn = an+1 / an How do we prove that.. rn = 2 - 1/(rn-2 + 1) for all n>2...

Homework Statement What is the limit of { \frac{n+1}{2n} } as n --> oo. Prove your answer. The Attempt at a Solution This is example from my book. Here is the problem: *I am using the capital letter "E" instead of "ε" in latex-code. Intuitively, 1 is small relative to n as n gets large, so...

erm just a quick question, as i read something that didnt quite make sense to me anyway, our sun will stay in its current form for about another 5 billion years before inflating into a red giant, now i was just browsing wikipedia reading up about various things, and was reading the article about...

Just a check of my work please, The topic is series and sequence, Question: The sequence u1, u2, u3,... where u1 is defined by u1=2 and Un+1 =un+4 Find the nth term, un, of the sequence. I got the answer, u1=-1 u2=-1+4= 3 u3= 3+4 = 7 So: un = 4(n-1)-1 This seems to work but not sure if it...

Homework Statement Corey has take a job with an initial salary of $26,000 an annual raises of $1,250/ (a) What will his salary be in the 6th year? (b) How much money in total will Corey have earned after six years? Homework Equations an = a1 + (n - 1) d Sn = n / 2 (a1 + an)...

Homework Statement For the arithmetic sequence (2 - x), (-6 + 2x), (x + 2), solve for x and find t10. Homework Equations an = a1 + (n - 1) d The Attempt at a Solution Would I have to start off like this below::: an = a1 + (n - 1) d d = (-6 + 2x)-(2 - x) = (x + 2)-(-6 + 2x)

Find \sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{1+...}}}}

Bench Top's thread revived a number string curiosity that once stumped me. I wonder if anyone else saw anything like this before and can give the sequence a name. Description 1. The sequence comprises only numbers from 1 to 2n. 2. Each number from 1 to 2n appears once and only once. 3...

Question : An infinite sequence of independent trails is to be performed . Each trails resulting in a success with probability p and failure with probability 1-p . What is the probability that i) atleast 1 success occurs in the first n trails ; ii) exactly k success occur in the first n...

Let (xn) be a seq of real nos and let sn = x1+x2+x3+...+xn / n. prove that if if xn is bounded and monotone, then sn is also bdd and monotone. How can i got about this one.. ? I got it in the test today and i couldn't figure it out. only hint i could think of is how do i prove if xn...

Homework Statement Let x1 > 9000, and xn+1 = )2009xn + 2010)/2011 for n >1 show that (xn) converges and find its limit Homework Equations Definition of a limit, Monotone Convergence Theorem. The Attempt at a Solution Since xn+1 is monotone for n>1 and bounded, then it...

Homework Statement Given a sequence of rational numbers, defined inductively as follows: s1 = 1, sn+1 = sn/2 + 1/sn, n>=1 prove that 1<=sn<=2 forall n>=1 Homework Equations The Attempt at a Solution I've got the solution to this but I don't understand a certain part, I was...

I'm reviewing for my analysis exam, and am having trouble with this question: Let (fn) be the Fibonacci sequence and let xn=fn+1/fn. Given that lim (xn) = L exists, find L. I believe I know what technique I have to use. I think I have to find two subsequences of (xn), write one in terms of the...