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

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46 views

A divergent sequence of integrals

Let {$s_{n}$} be a sequence of increasing step functions that converges pointwise to a limit function $f$ on an interval $I$ that is unbounded. $f(x)\ge 1$ almost everywhere on $I$ . ...
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0answers
17 views

Sum of Bell Polynomials of the Second Kind

A problem of interest that has come up for me recently is solving the following $$\frac{d^{n}}{dt^{n}}e^{g(t)}$$ There is a formula for a general $n$-th order derivative of a composition as shown ...
2
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1answer
50 views

Prove that $a_j=b_j$ for the $2$ sequences $a$ and $b$

Let $n$ is a natural number and $(n,6)=1$. Given $2$ sequences $a$ and $b$ such that $a_1>a_2>\ldots a_n$ and $b_1>b_2>\ldots b_n$. And for all $1 \leq j < k <l \leq n$, it is ...
6
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3answers
71 views

sequence of primes in arithmetic progression

The question is: Suppose $p_1<p_2<...<p_{15}$ is a sequence of prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$. Let ...
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2answers
51 views

Third axiom of Kolmogorov axioms

Let us define for a countably infinite set $S$ of real numbers that can be enumerated as $x_1,x_2,\cdots$, $$P(S) = \sum_{x \in S}p(x) = \sum_{i=1}^\infty p(x_i) = \lim_{n \to \infty}\sum_{i=1}^n ...
1
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1answer
32 views

Alternating Series Test Conditions

I'm learning about the Alternating Series Test which can be found here. My question is: Can you give an example of where $\lim_{n\to\infty} b_{n} = 0$ but $b_{n}$ is an increasing sequence?
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2answers
35 views

Limit and limit points

What is the basic difference between limit and limit points, and if a sequence has one unique limit how it can have a lot of limit points
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0answers
27 views

Does the Borel-transform of the Lerch-Transcendent have a name/simple expression?

The Lerch-transcendent as given in Mathworld is $$ \Phi(z,s,a)= \sum_{k=0}^\infty {z^k\over (a+k)^s}$$ I'm fiddling with series of the form $$ f_n(z)=\sum_{k=0}^\infty {z^k\over (1+k)^n} $$ and their ...
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0answers
33 views

how to solve for a term in the denominator of a summation? [closed]

For example, if I have: $$\displaystyle \sum_{i=1}^n \frac{1}{t_i + x} = \cdots$$ is there a way I can extract $x$ out of the summation?
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0answers
9 views

Flat frequency domain response

If I have a certain sequence having a flat frequency domain response, is its time domain autocorrelation a delta? In other terms, is a flat frequency response necessary and sufficient condition for ...
1
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2answers
57 views

Proving a theorem about Fourier coefficients

I need to prove this: Let $f$ be a $C^1$ function on $[-\pi, \pi]$. Prove that the Fourier coefficients of $f$ satisfy $|a_n| \leq \frac{K}{n}$ for some constant $K$. Can someone please let me ...
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4answers
382 views

Sum function of a series

Does anyone know what is the sum function $f(x)$ of the series $\displaystyle\sum_{n=1}^\infty \frac{\cos(nx)}{n^2}$? I have no idea how to find a sum function... Any help would be appreciated.
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1answer
16 views

Coefficients of general Fourier Series

I know how to compute coefficients of Fourier Series on an interval of $2\pi$. But in this case I need to find the sine series of $f(x)=b$ on the interval $x \in [-L,L]$. Can someone please let me ...
0
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1answer
17 views

Does the series $\displaystyle\sum_{j=1}^\infty -\log (1-p_j^{-3/4})$ diverge, where $\{p_j\}$ is the set of primes in increasing order?

Here, $\log$ is the natual logarithm. Is there a simple convergence test I can use? Thanks.
3
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0answers
100 views

Using a visual “proof” to show that $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n =1$

The "proof without words" that $\sum_{n=1}^{\infty} \left(\frac 12 \right)^n =1$ is fairly well known: But why can't we apply the exact same logic to $\sum_{n=1}^{\infty} \left(\frac 34 \right)^n$ ...
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1answer
18 views

Sketching the graph of a sequence of functions [closed]

Does anyone know how to sketch graphs of a sequence of functions? Say, I need to sketch $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ for $x \in [0, 2]$. But I have no idea where to start. Can someone please help? ...
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0answers
42 views

Show that the series does not converge uniformly on $\mathbb{R}$

My Work: Here, according to the given facts $f(0)=0$ and $f$ is strictly increasing. I proved part (a) and (b) but failed to prove (c). I was going to use the definition (actually wanted to show ...
2
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0answers
28 views

Order of Magnitude

It is well known (see here, for example) that we have $$ \psi\left(\frac{1}{2},T\right)=\sum_{p\leq T}\frac{1}{p}=\log(\log T)+A+O\left(\frac{1}{\log T}\right), $$ where $\psi(\sigma,T)=\sum_{p\leq ...
2
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1answer
59 views

Is there a divergent series with “largest” terms?

Suppose $a_n >0$ and $\sum_{n=1}^{\infty}a_n$ converges. Define $$r_n = \sum_{k=n}^{\infty}a_k$$ Does $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverge? My thinking is yes. Could someone give ...
4
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2answers
129 views

Convergence of sequence: $f(n+1) = f(n) + \frac{f(n)^2}{n(n+1)}$

I am trying to work out a problem on some old analysis qual exam, and managed to reduce it to this question, but I can't seem to figure out this final step: Consider the sequence defined by the ...
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2answers
97 views

Find a closed form of the series $\sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2}$

The question I've been given is this: Using both sides of this equation: $$ \frac{x}{1-x} = \sum_{n=1}^{\infty}x^n $$ Find an expression for: $$ \sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2} $$ Any help ...
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3answers
185 views

Formulae for sequences

Given that for $1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}$ deduce that $(n+1)^3 + (n+2)^3 +\cdots+ (2n)^3 = \frac{n^2(3n+1)(5n+3)}{4}$ So far: the sequence $(n+1)^3 + (n+2)^3 +\cdots+ ...
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2answers
75 views

Does anyone understand this proof: If $A$ is closed and bounded $\implies A$ is sequentially compact.

This is how it goes, I will highlight the parts in yellow which I don;t understand why it is , or the idea behind it. $A$ is bounded so $(\forall x \in A)(\exists M > 0)(\|x\|<M)$ Let ...
1
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1answer
25 views

Bernoulli-like generating function

What are the coefficients of the series for: $$\frac x{e^x+1}$$ It looks similar to the Bernoulli generating function, but the $+$ sign is throwing me off. I already found the series for its ...
0
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1answer
19 views

Possible value of $x$ so that fractions are in simplest form.

Which of the following could be the possible value of $x$ for which each of the fractions is in its simplest form, where $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to ...
0
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2answers
37 views

convergence of the series for $p>0$ , $\sum n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt n}\right)$

Discuss the convergence of the series for $p>0$ , $$\sum n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt n}\right)$$ I tried through ratio test. But it fails. I think it will be by comparison ...
0
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1answer
19 views

Combinatoric for number of ways to have monotone-increasing sequence

I hope I am using the right term. By monotone-increasing I mean to imply that it is a non-decreasing sequence. So for example a sequence $1, 1, 2, 5, 6, 10, 10, 11$, etc. Anyhow, consider a ...
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1answer
25 views

Applying squeeze theorem to conditionally convergent series

Suppose that the series ∑_n≥1(a_n) converges conditionally. Then by the Riemann Series theorem, for any real number L there exists a rearrangement of a_n(let's call it b_n) that converges to L. For a ...
2
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1answer
229 views

Partial sum of exponential series strictly increases after certain step

While trying to show that partial exponential series evaluated at two different values are strictly increasing provided that sufficient number of terms are applied I stuck at a problem. Given two ...
0
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1answer
62 views

Convergence of the series $\sum_{n=0}^\infty \sin(n! \pi m \sin(1))$

In this exercise I was asked to prove the convergence of the following infinite sum: $$\sum_{n=0}^\infty \sin(n! \pi m \sin(1)),$$ where $m$ denotes any integer. I don't have any idea on how to ...
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0answers
28 views

Is this :$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $ irrational series for every natural number $k$?

Is this: $$\sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!} $$ irrational series for every natural number $k$? Where : $\sigma_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of ...
1
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1answer
30 views

General methods of solving non-linear recurrences

Let's consider a sequence $$ \{a_{n} \} _{n=0}^{\infty}, a_{0}=1, a_{n+1}=\frac{a_{n}}{1+na_{n}}$$. Taking $$b_{n}=\frac{1}{a_{n}},$$ we bring the exact reccurence to the following one: $$b_{0}=1, ...
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3answers
90 views

Show that $x_n$ is convergent.

My Try: It is clear that $x_n$ is monotonically increasing. If we assume that the sequence converges to $a$ then $\displaystyle a=a+\frac{\sqrt{|a|}}{n^2}$. Hence $a=0$. So, I was going to prove ...
2
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1answer
29 views

Convergence of a sequence of $2\times 2$ real matrices

My Try: So $a_n$ can be written as a series very similar to the taylor series of sin: $\displaystyle a_n=\sum_{k=0}^n \frac{(-1)^k b_k}{(2k+1)!}$ for some $b_k$ to be determined. But it is very ...
0
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1answer
43 views

Evaluating a cube root

How to evaluate $(8.024)^{1/3}$ from $(1+3x)^{1/3}$.I already expand it until $x^3$ but i still can't get the answer. I tried googling for the working using binomial theorem but i failed.
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1answer
51 views

Ratio in sequences and series

In a geometric series where the common ratio is $r$ ($r^2$ is not $1$) the sum of the first $13$ terms is three times the sum of the first $6$ terms.How do I find in any order the ratio of the sum of ...
2
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2answers
56 views

Convergence of an oscillating recursive sequence

Define the recursive sequence $ q_{n+1} = \dfrac{q_n+2}{q_n+1};\;q_0=1 $ If we knew that $ q_n \to q;\;n\to \infty $ then it's easy to show what follows $ q_{n+1}\left(q_n+1\right) = q_n+2 $ $ ...
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1answer
21 views

Convgergence of series with different parametes [closed]

I have this series : $$\sum\limits_{n=1}^{\infty} n^{-a}\log(n)^{-b}$$ For what values of $a$ and $b$ does the series converge?
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2answers
60 views

Does this series converge or diverge and by which test?

$$\sum_{n=1}^\infty (-1)^{n+1} \sin(1/n^3)$$ I tried to apply the divergence test. I know $\lim_{n\to \infty}$ is 0 for $b_n$ but I don't think $b_n$ is decreasing. any ideas on how I can test this ...
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0answers
50 views

Can this sum be exactly calculated or be converted into an integral? [closed]

$\sum \limits_{n = -\infty}^\infty \cos\sqrt{a n^2 + bn + c}$, where $a, \ b$ and $c$ are arbitrary constants, $n \in \mathbb Z $, integers.
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1answer
82 views

How to show that his series converges or diverges using LCT or CT?

$$\sum_{n=1}^{\infty}\left (\sqrt{n^4+1}-n^2\right)$$ The question states that either the limit comparison or comparison test can be used to determine whether the series converge or diverge. I tried ...
0
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1answer
15 views

Separability of $l^{p}$ spaces

How can I prove that the space $l^{p}$ equipped with the norm (for $x=(x_{n}) \in l^{p})$: $||x||_{p}=(\displaystyle\sum_{n}|x_{n}|^{p})^{1/p}$ Is a separable space? (i.e. showing that there is a ...
1
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1answer
45 views

Why doesn't this work for Rudin Exercise 8 Chapter 3 series proof?

Okay so Here is the problem: If $\sum{a_n}$ converges and $b_n$ is bounded and monotonic, prove that $\sum{a_nb_n}$ converges. So I can follow the long epsilon based proof and I'm good with all ...
5
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2answers
107 views

Determine all real $x$ for which the series $\sum\limits_{k=1}^\infty\frac{k^k}{k!}x^k$ converges.

Determine all real $x$ for which the following series converges: $$\sum_{k=1}^\infty\frac{k^k}{k!}x^k.$$ You may use the fact that $$\lim_{k\to\infty}\frac{k!}{\sqrt{2\pi k}(k/e)^k}=1.$$ ...
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2answers
78 views

How to find the limit of a sequence?

Question: If $0 < x < \frac{\pi}{2}$ and $f_k(x) = \tan(x)+\frac{1}{2}\tan(x/2)+ ...+\frac{1}{2^k}\tan(x/2^k)$. In Sigma Notation: $$f_k(x) = \sum_{n=0}^k \frac{1}{2^n}\tan\frac{x}{2^n}$$ ...
0
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2answers
73 views

Test the convergence of the series $\sum_{n=1}^{\infty}a_n$

Test the convergence of the series $\sum_{n=1}^{\infty}a_n$ , where $$a_n=\begin{cases}\dfrac{1}{n^2} & \text{ if $n$ is not a square integer},\\[6pt] \dfrac{1}{n^{2/3}} & \text{ if ...
3
votes
4answers
401 views

1+4+10+20+35+…=? [closed]

Is there a finite value to the infinite sum of all the tetrahedral numbers: $$\sum_{n=1}^\infty \frac{n(n+1)(n+2)}{6}.$$ I know it's a divergent series, but I hear that $$ ...
2
votes
1answer
114 views

Compute this integral

$$ \displaystyle \int_{0}^{\infty}{\frac{{log}^{2}(1-{e}^{-x}){x}^{5}}{{e}^{x}-1} dx} $$ What I have done - $ \displaystyle I(k) = \int_{0}^{\infty}{\frac{{x}^{5}}{{e}^{x}{(1-{e}^{-x})}^{k}}}$ ...
3
votes
2answers
106 views

The converges of $ \sqrt { 2 } +\sqrt { 2-\sqrt { 2 } } +\sqrt { 2-\sqrt { 2+\sqrt { 2 } } } + …=$

I would like to know wheather this series converge or diverge? $\sqrt { 2 } +\sqrt { 2-\sqrt { 2 } } +\sqrt { 2-\sqrt { 2+\sqrt { 2 } } } +\sqrt { 2-\sqrt { 2+\sqrt { 2+\sqrt { 2 } } } } ...
1
vote
4answers
71 views

Sum of roots of unity, proving that $1+w+w^2…+w^{n-1}=0$ [closed]

If $w$ is a unit square of rank $n$ (meaning $w^n=1$), s.t $w$ is not $1$. Prove that $1+w+w^2.....+w^{n-1}=0$. We're pretty sure that we need to use induction, its easy to prove for $n=2$ but ...