Sequences Definition and 590 Threads

I just can't get the following question. Can someone help me out? Q. Let a < a_1 < b_1 and define a_{n + 1} = \sqrt {a_n b_n } ,b_{n + 1} = \frac{{a_n + b_n }}{2} . a) Prove that a_n \le a_{n + 1} \le b_{n + 1} \le b_n for all n. b) Deduce that the sequences {a_n} and {b_n} both...

I have to use the partial fraction technique on 1/(4k^2 - 1)... ANSWER: So far so good and I get 1 / 2(2k-1) - 1 / 2(2k+1), is this correct? I now need to show that ? \sum 1 / 4k^2 - 1 = n / 2n + 1 Please help :confused:

For a sequence a_1, a_2, ... in R^n to be convergent there are (at least) 2 theorems, as follows: if for all epsilon>0 there exists an M such that when m>M, then |a_m-a|<epsilon and also: If u(epsilon) is a function such that u(epsilon)-->0 as epsilon-->0, then the sequence is...

my problem is regarding sequences: Sum first 20 terms of a=r^k-1 terms are 2, 4/3, 8/9,... and ratio is 2/3 and r < 1 so its an inflated series and 2(2/3)^19 = .009021859795 BUT it is an inflated series right? and my...

I think I'm having some trouble on this. First I'll state what the question is then I'll show what i have and my reasoning. Determine if the sequence{b sub(n)} is convergent by deciding on monotonicity and boundness. given: b sub(n)=n^2/2^n First I plugged in numbers for n starting with...

Does anyone know what are the uses of Sequences and Series in real life:confused:

Question Consider the sequence \{p^n\}_{n\in\mathbb{N}}. Prove that this sequence is Cauchy with respect to the p-adic metric on \mathbb{Q}. What is the limit of the sequence?

Let (x(n)) and (y(n)) be sequences of positive numbers such that lim(x(n)/y(n)) = 0. If lim(x(n)) = +∞, then lim(y(n)) = +∞ If (y(n)) is bounded, then lim(x(n)) = 0 To me this is self-evident. But HOW can it be proved?

Hey there everyone, We were discussing factorial sequences in my last pre-calculus class. Factorials are pretty cool. I asked if they had any rel world applications or examples I could put into my notes. She then told us if we could find an example that we'd get extra credit on our quiz, I'm...

I need to determine whether Sigma [sin(1/x)] for x=1 to x=infinity converges or diverges. I have a feeling that it diverges, but I don't know how to prove it. Thanks

hello all been workin on this problem: let An Bn and Cn be sequences satisfying An<=Bn<=Cn for all n an element of the natural numbers suppose that An->x and Cn->x, where x is a real number show that Bn->x this is how i did it A_n\le B_n\le C_n \forall n\epsilon N...

hello all I found this rather interesting suppose that a sequence {x_{n}} satisfies |x_{n+1}-x_{n}|<\frac{1}{n+1} \forall n\epsilon N how couldn't the sequence {x_{n}} not be cauchy? I tried to think of some examples to disprove it but i didnt achieve anything doing that, please...

Definition: Suppose T is an index set and for each t in T, X_t is a non-void set. Then the product \Pi_{t \in T}X_t is the collection of all "sequences" \{x_t\}_{t \in T} = \{x_t\} where x_t \in X_t. Does this mean that \Pi_{t\inT}X_t is the set containing all possible sequences defined by...

I wonder do the genes have the same exon-intron junction sequences or do they have different junction sequences? I was told that all genes have this general junction sequences of the exon-intron-exon: 5'---exon---A/CG-><-GUPuAGU----intron-----Py12NPyAG-><-G---exon---3' The arrows indicate...

hey... I'd appreciate it if you could verify my answers.. Q1) Write a rule for the nth term of the arithmetic sequence 1,6,11,16... Then find a10. A1) An=A1 + (n-1)d An=1+ (n-1)5 A10=1+(10-1)5 A10=1+45 A10=46...

• Why should some Poly-A sequences lay upstream of a reporter gene? • Isogenic cell lines? • Null cell lines? Any ideas are greatly appreciated.

Im in need of some guidance. No answers, just guidance. :smile: Question. Let (x_m) be a Cauchy sequence in an inner product space, show that \left\{\|x_n\|:n=1,\dots,\infty\right\} is bounded. proof From the definition we know that all convergent sequences are Cauchy...

Hello, could someone please help me with the following questions? Q. Determine which one of the following sequences converge and which do not converge. Explain your answers. For any sequence that converges, find the limit. (i) \frac{{n + \left( { - 1} \right)^n }}{{2 + \left( { - 1}...

Hello all: Given a_n = \frac {n}{\alpha^n} and \alpha is a number greater than 1, we assert as n increases the sequence of numbers a_n = \frac {n}{\alpha^n} tends to the limit 0. Let us consider the sequence \sqrt a_n = \frac {\sqrt n}{(\sqrt \alpha)^n} We put \sqrt \alpha = 1+h...

Consider x_1,y_1 \in \mathbb{R} such that x_1>y_1>0 and \{x_n\},\{y_n\} the two sequences defined for all naturals by x_{n+1}=\frac{x_n+y_n}{2}, \ \ \ \ \ y_{n+1}=\sqrt{x_n y_n} Show that the sequence \{y_n\} is increasing and as x_1 for an upper bound. I would appreciate some help on...

I'm trying to find out tests with regards to determining if a limit of a sequence exists or not (ie convergence of sequences), since evaluating a particular limit may not always possible. For example it seems to me that if for a particular sequence a, if limn->infty a(n+1)/a(n) = 1, then...

Consider the following statement: If \left\{ a_n \right\} and \left\{ b_n \right\} are divergent, then \left\{ a_n b_n \right\} is divergent. I need to decide whether it is true or false, and explain why. The real problem is that I checked the answer in my book; it's false, but I...

Sequences a_n and b_n are defined in the follwing way: a_1=x; b_1=y; where 0<x<y and: a_(n+1) = (a_n+b_n)/2 b_(n+1) = sqrt(a_(n+1)+b_n) Proof, that both sequences are convergent to the same limit and find this limit. Thanks a lot for any help.

I am teaching honors calculus in college, and trying to teach something about convergence of sequences and series. my class has apparently never seen a genuine proof in high school and have no idea how to begin one (answer: with the definition). I have had students ask me what "QED" stands...

Analysis problem (sequences)--please help Here is the definition: t_n = [s_1 + s_2 + ... + s_n] / n ; n >/= 1 I have to show that if lim n-> [infinity] s_n = s, then lim n-> [infinity] t_n = s First of all, I don't think it's true. Because if s is finite, then lim s/n as n-> [infinity]...

Sequences and series... My textbook says that a progression is another name for a series, but the dictionary says it is another name for a sequence - which is it?

Series, Sequences and Progressions... My textbook says that a progression is another name for a series, but the dictionary says it is another name for a sequence - which is it? Thanks.

Let {an}(n goes from 1 to infinity) be a sequence. For each n define: sn=Summation(j=1 to n) of aj tn=Summation(j=1 to n) of the absolute value of aj. Prove that if {tn}(n goes from 1 to infinity) is a Cauchy sequence, then so is {sn}(n goes from 1 to infinity). I started this...

If a general statement like an->a where (an) is a sequence of non negative real numbers, how would we prove the sqare root an->the square root of a. When a=0, this can easily be done. But I don't see how this is possible from the given information for the case where a>0. Thanks for any help...

where can we get the full length of amino acid sequences? i have very little bioinformatic background so please the more you explain the better it would be :-p :biggrin: thank you!

hello everybody! i have to find match between two different peptide sequences (looking for related sequences), but don't know where and how i can do it. i have very little bioinformatic background. i hope for some help. thanks alot!

How do I prove by mathematical induction that the sequence given by a_{n+1}=3-\frac{1}{a_n} \qquad a_1=1 is increasing? The difficulty in finding it myself is that recursive sequences are not familiar to me---i.e. usually, I am able do the following steps without a problem: (A)...

Problem: (a) Let a_1 = a, a_2 = f(a), a_3 = f(a_2) = f(f(a)), \ldots, a_{n+1} = f(a_n), where f is a continuous function. If \lim _{n \to \infty} = L, show that f(L) = L . (b) Illustrate part (a) by taking f(x) = \cos x , a = 1, and estimating the value of L to five decimal places...

Hello all Let us say we are given a sequence of order 2. By order 2 I mean that we have a sequence in which the differences between the terms forms a sequence of order 1, which has a constant difference between terms. How can I prove that the nth term of a sequence of order 2 can be...

Hi... 1. so can i say that a recurrence relation is a description of the operation(s) involved in a sequence...?... 2. is the formula for an arithmetic sequence, a recurrence relation...?... and is the formula for a geometric sequence, a recurrence relation...?...

How do i do a Multiple Alignment with 4000 sequences. offline version of ClustalW hangs online version accepts only 500 sequences What is the maximum number of sequences that can be given to CluatalW ? Any other gud softw . PLease let me know ...

Is there some concise mathematical form to express the fact that a sequence repeats with period t beginning with the nth term? For example, the sequence {1,2,6,3,7,3,1,7,3,1,7,3,1,7,3,1,...} repeats with period 3 beginning with the 5th term. Can we say, for all n>4, if b=an then b=an+3? I...

Can anybody help me solving this? Write in terms of factorials n((n^2)-1) The correct answer is (n+1)!/(n-2)! but I don't know how to get there, and since it's week- end I have no chance to ask anyone teachers, etc. //Martin

I'm having trouble with these type of probles (where a negative log comes up): (All of this is solving without sigma notation) Find the number of terms in these geometric sequences and the sum of the numbers. 11, -22, 44,...,704 I know that a1 = 11, r = -2, and an = 704, so I did...

I'm trying to get an A in honors AlgII/Trig and it is impossible, but I won't give up, so I have a few questions. I'm not sure how to find the first two terms of a sequence (I got a few right, but most wrong and I don't know what's wrong). One of the problems is: a5 = 20; a8 = 4/25. I set...