Tagged Questions

For questions about recurrence relations, convergence tests, and identifying sequences

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1
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2answers
33 views

Divergent to $\infty \Rightarrow$ Divergent?

In our lecture, we defined a sequence $\left(a_n\right)_{n\in\mathbb N}$ to be divergent if it does not converge, and additionally to be divergent to $\pm \infty$, iff: $$\forall \epsilon \in \mathbb ...
2
votes
3answers
53 views

$\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$

How to find the sum of the following series: $\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$ Any hints.
0
votes
1answer
16 views

Demonstrate series of Maclaurin

Find the Maclaurin series of $$f(x)=xe^x$$ Integrate this series term by term in the closed interval $[0,1]$ and demonstrate that: $$\sum^\infty_{2} \frac{1}{(n-2){} !n} = 1$$ I tried it: ...
1
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0answers
31 views

Finding the convergent value

The sequence is defined as follows : Start : $(x_0,y_0)$ with $ 0 < x_0 < y_0 $ Step : $x_{n+1} = \frac {x_n+y_n} {2}$ , $y_{n+1}= \sqrt{x_{n+1}y_n} $ Find $\lim_{n\to \infty}(x_n,y_n)$ . ...
0
votes
3answers
26 views

Functions of sequences and convergence

(a) If $f$ is continuous on $[0,\infty)$ and {$x_n$} is a sequence in $(0,\infty)$ such that {$f(x_n)$} diverges to $\infty$, then $\lim_{n \to \infty} x_n = \infty$. (b) If $f$ is continuous on ...
0
votes
1answer
16 views

Alternating Series Test for Divergence

Alternating Series Test if the alternating series $$\sum^\infty_{n=1}(-1)^{n-1}b_n=b_1-b_2+b_3-b_4+b_5+...\;\;\;\;\;b_n\gt0$$ satisfies $$(\text{i)}\;\;b_{n+1}\leq b_n\;\;\;\;\;\text{for all ...
0
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2answers
21 views

continuity and sequences

If $f$ is continuous on $[a,b]$ and {${x_n}$} is a sequence in $(a,b)$, then {$f$(${x_n}$)} has a convergent subsequence. True or False? If true, prove. If false, give a counterexample. I'm guessing ...
0
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0answers
8 views

$p_n(x)=a_nx^2+b_nx+c_n$,$a_n,b_n,c_n\in\mathbb{R}\forall n\ge 1$

$p_n(x)=a_nx^2+b_nx+c_n$ be a sequence of quadratic polynomials, $a_n,b_n,c_n\in\mathbb{R}\forall n\ge 1.\lambda_0,\lambda_1,\lambda_2$ are dstinct reals $\ni$ $\lim p_n(\lambda_0)=A_0,\lim ...
1
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2answers
52 views

Trying to find the closed form for the nth term of $\frac{1}{1-x^4}$

I know that $\frac{1}{1-x^4}$ is the generating function for the sequence (1, 0, 0, 0, 1, 0, 0, 0, 1, ...) I don't know how to find the closed form for the nth term though. Itried messing around with ...
0
votes
1answer
22 views

Discuss the convergence of the sequence who's nth term is given by…

$$a_n = \left(1-\frac{1}{2n}\right)^n$$ Please explain the process of how this is solve, I'm really confuse and struggling on how to figure out series and sequences. Since this is a sequence, is ...
0
votes
1answer
35 views

Prove that a series is convergent

I have a series which is as follows $$\sum_{n=1}^{\infty}\left(\left(1+\frac{1}{n^3}\right)^n-1\right)$$ and I am asked whether it converges or diverges. I think this series is convergent and I ...
10
votes
2answers
50 views

An equivalent for $\sum_{n=0}^{\infty} e^{-x\sqrt{n}}$ as $x$ tends to $0^+$

I would like to obtain an equivalent form for $$ f(x)=\sum_{n=0}^{\infty} e^{-x\sqrt{n}} $$ as $x \rightarrow 0^+$. I tried without success to "remove" the $\sqrt{\cdot}$ in the summand by summing ...
2
votes
2answers
87 views

Arithmetic progressions.

"Consider an 4 term arithmetic sequence. The difference is 4, and the product of all four terms is 585. Write the progression". My way of finding the progression seems like it will take too long, but ...
0
votes
1answer
29 views

Voltage Distribution Inside a Cylinder [on hold]

I was assigned this problem, and quite honestly I do not know where to begin. If I could get some help and an explanation of the Bessel function, also? Thank you. I know my conditions are: ...
0
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0answers
14 views

Prove series converge using comparison test

I got a question which is Suppose that $a_n \ge 0 $ $\forall n \in \mathbb{N}$ and that $\sum^{\infty}_{n=1}a_n$ converges. Prove that $\sum^{\infty}_{n=1}(a_n)^2$ also converges. And what I did is ...
1
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1answer
28 views

prove that $n^\epsilon$/log(n) goes to infinity without derivatives or functions

I'm looking for a way to prove that for every $\displaystyle{\quad\epsilon\ >\ 0\,,\quad{n^{\epsilon} \over \log\left(\, n\,\right)} \to \infty,\quad}$ treating it only as a sequence, without using ...
0
votes
2answers
24 views

A sequence $(a_n)$ where $\exists M>0$ such that $\forall n\in\mathbb{N}$, $\sum\limits_{k=1} ^n |a_{k+1}-a_k|\leq M$. Show $(a_n)$ is Cauchy sequence

Question: Let $(a_n)_{n\in\mathbb{N}}$ be a sequence where there exists $M>0$ such that for all $n\in\mathbb{N}$, $\sum\limits_{k=1} ^n |a_{k+1}-a_k|\leq M$. Show that $(a_n)_{n\in\mathbb{N}}$ ...
-2
votes
1answer
40 views

Guess the missing number [on hold]

What number is missing from the sequence? 1, 2, 3, 5, 7, 11, 15, 22, 30, *, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627
-1
votes
3answers
25 views

Proving zero raised to the zeroth power is equal to one using the power series?

Its been a while since I have done math, so I wondering if there was a proof for this, if so what? Is zero to the zero power 1? Or is it not defined?. Thank you
1
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1answer
36 views

Series convergence and limit

I am trying to solve an exercise that asks me this: Prove using the $ε–N$ method that the sequence $a(n) = \frac{n^2 + n - 1}{n^2 + n}$ converges and state the limit. My attempt is the following: ...
-2
votes
1answer
33 views

Evaluating the series with arctangents: $\sum_{r=1}^\infty \tan^{-1}\frac{2r}{2+r^2+r^4}$

If $$S=\sum\limits_{r=1}^\infty\tan^{-1}\left(\frac{2r}{2+r^2+r^4}\right)$$ then what is cot S? Options: A) 1; B) 3; C) 1/3; D) 2 Does it converge? I don't really know how to find the ...
1
vote
1answer
46 views

Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge? [on hold]

Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge? $a_{2n}$ is a subsequence of $a_n$ and $a_{3n},a_{2n}$ are subsequences of $a_{6n}$ so they all have the same limit But what ...
0
votes
0answers
21 views

properties of partial sums of series $\sum \sin(n)$

There have been many questions posted here about the sequence $s(x)_n = \sin(nx)$ and its corresponding series $S(x)$. In particular, it has been shown that $s(x)_n$ only converges if $x$ is a ...
0
votes
1answer
45 views

How to evaluate the following sums: [on hold]

How to evaluate the following sums: $A.1+\frac {3}{4}+\frac {3.5}{4.8}+\frac {3.5.7} {4.8.12}+...$ $B.\frac {1}{3}+\frac {1}{4}.\frac {1}{2!}+\frac {1}{5}.\frac {1}{3!}+....$ Is there any special ...
9
votes
1answer
103 views

Does $\sum |\sin n| / n$ converge? [duplicate]

How can I prove if the following series converges? $$\sum_{n\geqslant1} \frac{|\sin n|}{n}$$ I can't use differential or integral calculus. I've tried using Dirichlet and Cauchy tests, but they ...
0
votes
2answers
50 views

If $a_{2n}$ and $a_{3n}$ converges does $\lim(a_{2n}) = \lim(a_{3n})$

If $a_{2n}$ and $a_{3n}$ converges does $\lim(a_{2n}) = \lim(a_{3n})$? I tried ${\sim}\infty$ possibilities but couldnt find a counter example so I tend to believe it is true?
1
vote
2answers
29 views

Proving the convergence of this series

I have the series $\displaystyle\sum_{n=1}^{\infty}{\frac{6\sqrt{n}+5}{2n^2-n}}$ I am sure that this series converges, but I need to prove this and would like to use the comparison test to do so. ...
0
votes
1answer
18 views

If $x_k ≥ 0\;\forall \in \mathbb N$, and $y_k$ a bounded sequence, then the series $\sum_{k=1}^\infty x_ky_k$ converges

Hi I'm really struggling with this proof. For a start I'm struggling to believe it's true: For example, if we take $x_k = \dfrac{1}{k^2}$ and $y_k = -k^3$ (which is bounded above by any positive ...
1
vote
1answer
41 views

Definite integral of general polylogarithm

$$\int_{0}^{1} Li_k(x) dx$$ $$Li_k(x) = \sum_{n=1}^{\infty} \frac{x^n}{n^k}$$ $$\int_{0}^{1} Li_k(x) \,dx = \sum_{n=1}^{\infty} \int_{0}^{1} \frac{x^n}{n^k} \,dx$$ From Fubini's theorem, I suppose ...
1
vote
4answers
51 views

Sum of all triangle numbers

Does anyone know the sum of all triangle numbers? I.e 1+3+6+10+15+21... I've tried everything, but it might help you if I tell you one useful discovery I've made: I know that the sum of ...
0
votes
3answers
28 views

“The limit of a sequence is insensitive to finite changes in the sequence” - help me understand this sentence!

The context is p59 of "A primer on Hilbert Space Theory, by Alabiso and Weiss, where the example is given of a quotient $c_{00}/c_0$ which is "the quotient of the space $c_0$, the space of all ...
2
votes
3answers
36 views

Show that $\sum_{k=1}^\infty (a_k - a_{k+1}) = a_1 - l$ if $\lim_{k \to \infty} a_k = l$

Show that $$\sum_{k=1}^\infty (a_k - a_{k+1}) = a_1 - l$$ if $$\lim_{k \to \infty} a_k = l$$ Can anyone please provide me with hints as to go about solving this problem?
0
votes
1answer
25 views

Finding a function greater or less than factorial function

suppose we are given the sequence: $a_n = (-1)^n\frac{1}{n!}$ using squeeze theorem find the limit: $$\lim_{n\to \infty} (-1)^n\frac{1}{n!}$$ using the squeeze theorem. For factorials, $a_n$ how ...
0
votes
0answers
27 views

How many 4 digits prime numbers can be formed from 0,1,…,9 without repeated digits?

I'm just curious about the prime numbers in combinatorics. Can we use the combinatorics rule to find the number of prime number from given number, for example from the above condition? My attempt: I ...
1
vote
1answer
37 views

Nature of the series $\sum_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$

Does $\sum_{n=3}^{\infty} \dfrac{1}{(\log\log n)^{\log n}}$ converge or diverge ? I tried some tests , but nothing conclusive is coming . Pleas help
1
vote
1answer
21 views

Two definitions - do they differ?

I have the following definitions which may be true or false but for me it seems like both true A) If ($a_{2n} - a_n$) converges to $0$ then $a_n$ converges. B) If $a_n$ converges then ($a_{2n} - ...
2
votes
1answer
33 views

Summing the sequence $a(n) = \sin(n x) \exp(-nt)$

Consider the sequence $a(n)$ defined by $a(n) = \sin(n x) \exp(-nt)$, where $n = 0, 1, 2, 3, 4, \ldots$. The parameter $x$ is a real number. Parameter $t$ is a positive real number. It is clear that ...
0
votes
0answers
41 views

The convergence of the multiplication of two convergent series? [on hold]

If we know that \begin{equation} {\sum\limits_{n=1}^{\infty} }a_n \end{equation} and \begin{equation} {\sum\limits_{n=1}^{\infty} }b_n \end{equation} are convergent What about their ...
2
votes
3answers
105 views

Sum of convergent and non-convergent series, does it converge? And how to prove?

Series $a_n$ is convergent and $b_n$ is not-convergent. Will the sum $a_n + b_n$ converge? I think it will not converge, But how do I show it? I believe I have to use the definition. $|a_n - A| < ...
0
votes
3answers
45 views

How to do this limit?

$$\large\lim_{n\to \infty}\large\frac{\sum_{k=1}^n k^p}{n^{p+1}}$$ I'm stuck here because the sum is like this: $1^p+2^p+3^p+4^p+\cdots+n^p$. Any ideas?
2
votes
4answers
86 views

Is $\sum\frac{1}{\sqrt{n+1}}$ convergent or divergent?

$$\sum\frac{(-1)^n}{\sqrt{n+1}} \text{and} \sum\frac{1}{\sqrt{n+1}}$$ The first one is an alternating series, so it would just be: $$\sum (-1)^n\frac{1}{\sqrt{n+1}}\Rightarrow ...
0
votes
1answer
28 views

convergence of an alternating series without $a_n$

I have an infinite series that goes like: 3-$\frac{69}{5}$+$\frac{834}{25}$-$\frac{7734}{125}$+$\frac{62109}{625}$-$\frac{455859}{3125}$+$\ldots$ I can generate more terms of this series if needed. ...
1
vote
3answers
46 views

What is the difference between convergence of a sequence and convergence of a series?

I'm preparing for my calculus exam and I'm unsure how to approach the question: "Explain the difference between convergence of a sequence and convergence of a series?" I understand the following: ...
1
vote
1answer
52 views

Sequence in a hilbert space.

$M$ is a closed subspace of the Hilbert space $H$, and x $\in H$. Call $d = \inf_{y \in M} ||x - y||^2$ Show that there exist a sequence of elements $y_n$ of M such that $||y_n - x ||^2 \rightarrow ...
2
votes
2answers
22 views

Continuity and diverging sequences

Let $I = (0, ∞)$ and let $f : I → \mathbb{R}$ be a continuous and bounded funciton. Show that for any real number $S$ there exists a sequence $(x_n)$ such that $\lim x_n = ∞$ and $\lim (f(x_n + S) − ...
3
votes
0answers
51 views

A series from textbook

Does it make sense to try to compute it exactly since there is nothing that would suggest this is possible? $$\sum_{n=1}^{\infty} (-1)^{n-1}\dfrac{\log(n+1)}{n}$$
1
vote
2answers
89 views

Covergence test of $\sum_{n\geq 1}{\frac{|\sin n|}{n}}$

I need to prove that $$\sum_{n\geq 1}{\frac{|\sin n|}{n}}$$ is convergent. How should I do it?
0
votes
0answers
16 views

Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
0
votes
2answers
36 views

Proving convergence by proving a sequence does not diverge

Is it possible to prove convergence by proving that a sequence does not diverge? Especially I don't know how to deal with periodic sequences such as $\lim \limits_{n \to \infty } \sin(n)$.
1
vote
1answer
37 views

Can we find two sequences $(a_{n}),(b_{n})$ with several conditions

Can we find two sequences $(a_{n}),(b_{n})$ such that $$a_{n}>b_{n}>0 \\ \lim_{n→∞}a_{n}=\lim_{n→∞}b_{n}=0 \\ \lim_{n \to \infty}\frac{a_n^{-n}}{a_{n+1}^{-n-1}}<1<\lim_{n \to ...