Recurrence relations, convergence tests, identifying sequences

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Taylor series of $f(x)=\frac {e^x-1}{x}$

I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions. How to simplify the function so that it can be expanded more easily?
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1answer
40 views

Math arithmetic

So I got this question in my exam yesterday, and only a few people from my school could solve this. What's so hard about it? I have learned the simple arithmetic questions, but I don't really ...
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3answers
33 views

find the condition on A for the summation to be convergent

The summation is: $$\sum_{n=1}^\infty \frac{ \sqrt { n + 1 } - \sqrt n }{n^A}$$ I don't know how to even begin. Hints??
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0answers
38 views

funcitonal series convergence… SOS… [duplicate]

Is it possible to show that $f(x) = \sum_{k=0}^\infty (-1)^k \frac{x^k \sqrt{k}}{k!}$ converges even when $x\rightarrow \infty$ ? i.e. As a function of x, does $\lim_{x\to\infty} f(x)$ exist or at ...
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1answer
33 views

Alternating functional Series Convergence SOS…

Does the following series converge? $\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$ what is the radius of convergence?!!
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1answer
26 views

How do I write the generic series definition that can produce any term in the series expansion?

I'm trying to learn how to build the generic series definition for a series of numbers. For some reason I'm having a hard time pulling out this pattern. I'm always pulling out the wrong details for ...
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1answer
27 views

Extracting distinct sequence

Let $(x_n)$ be a non-constant sequence in $\mathbb{R}$ and $x_n\rightarrow p$ for some $p\in \mathbb{R}$. Can I always extract a subsequence $(x_{n_k})$ whose all elements are distinct? (of course ...
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1answer
159 views

Prove every integer exists in this sequence?

Please could someone give me a hint on this sequences question? The question is to prove that every integer appears infinitely many times in the following sequence: $$ \pm 1^{2} , \pm 1^{2} \pm 2^{2} ...
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2answers
36 views

“Translating” one value $- \infty$ to $+ \infty$ to another ($+ \infty$ to $ \gt 0$)

Well even if I need to use the following in a computer game this is a math question. I have a world map which I can scroll with a scroll velocity $(f(x))$ with my mouse. And I have a zoom factor $(x)$ ...
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0answers
22 views

Calculation the variance of the forecast error?

Hi there stuck on the following: Consider the model: $$y_{t}=(1+a)y_{t-1}-(a)y_{t-2}+\epsilon_{t}$$ where $\epsilon_{t}$ is a white noise problem: 1) Transform $y_t$ into some other series $w_t$ ...
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0answers
29 views

show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle [duplicate]

I need to show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle except the point $z=1$, well at $z=1$ we get our known divergent harmonic series, but I am not able to show easily ...
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2answers
39 views

Divergence of $\sum_{n=2}^\infty\frac{1}{(\ln n)^x}$ for $x>1$

How can we show that the series $$\sum_{n=2}^\infty\frac{1}{(\ln n)^x}$$ diverges for $x>1$ ? The book gives the following hint: consider $$\sum_{k=1}^\infty\sum_{n=M_{k-1}+1}^{M_k}\frac{1}{(\ln ...
3
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3answers
45 views

Value of series, Partialsum?

given is the following series $$\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$$ And I need to find its value. How can I start finding it? Thanks for all
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1answer
30 views

Monotonically decreasing sequence $(a_n) \to$ mo. dec. sequ. $(a_1+a_2+\cdots+a_n)/n$

I think this isn't quite difficult, however I don't get the point.. I have to prove: $(a_n)$ is a monotonically decreasing sequence. Show, that the sequence $\frac{a_1 + a_2 + \cdots + a_n}{n}$ is ...
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1answer
34 views

Relation between a sum of a series and the limit of a sequence

I'm stuck on this question Let $\{a_{n}\}$ a sequence of real numbers I need to show the series $\sum_{n=1}^{\infty}(a_{n} - a_{n-1})$ and the sequence $\{a_{n}\}$ are the same nature (convergent ...
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2answers
44 views

Soving Recurrence Relation

I have this relation $u_{n+1}=\frac{1}{3}u_{n} + 4$ and I need to express the general term $u_{n}$ in terms of $n$ and $u_{0}$. With partial sums I found this relation $u_{n}=\frac{1}{3^n}u_{0} + ...
3
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2answers
65 views

limit of series exponential

Compute the limit of the series $$\sum\limits_{n=4}^\infty 3\frac{2^{n+1}}{5^{n-2}}$$ How do you approach these types of problems? I'm thinking that this one is in indeterminate form, is that ...
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0answers
60 views

prove the numbers in a sequence are all square number

Given the sequence $\{y_n\}$ defined by: $$\begin{align*} &x_{n+1}=23x_n+2+y_n+2\\ &y_{n+1}=551x_n+24y_n+64 \end{align*}$$ for $n\in\Bbb N$ and $x_1=1,y_1=39$, prove that $x_n$ is a square ...
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1answer
50 views

convergence of series question

how do you determine if a series converges or diverges? Do you just look at their behavior?
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3answers
39 views

limits of sequences exponential and factorial

Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences: (a) $a_n=e^{5\cos((\pi/6)^n)}$ (b) $a_n=\frac{n!}{n^n}$ For part (a) do I just take the limit of the exponent part and ...
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4answers
34 views

Evaluation of a complex numbers partial sum

Let $w = e^{i\frac{2\pi}5}$. I would like to evaluate $$w^0 + w^1 + w^2 + w^3 +...+ w^{49}$$ Can anyone please give me an idea how to evaluate the expression? Thanks in advance
3
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0answers
78 views

Compute limit of the sequence $x_n$

Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq ...
3
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1answer
32 views

Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$

I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
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39 views

Solve $ z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots $.

I am trying to solve $z\in \mathbb{C}$ in terms of $a\in \mathbb{C}$, where $$ z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots. $$ I plugged $z= \sum_{k=0}^\infty c_k a^k $ into the ...
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2answers
47 views

Convergence of these series

$$\sum_{n=1}^\infty \frac{2nx^{n}}{(n+1)^{2}3^{n}} \tag{1}$$ $$\sum_{n=1}^\infty x^{n}\tan\frac{x}{4^{n}}\tag{2}$$ Is there any good article that describes an equivalents like if $$ ...
2
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0answers
35 views

Simplify series of exponentials

I would like to simplify the following series: 1.$$\sum_{n=1,odd}^{\infty}\frac{\exp(-an^2)}{(b-n^2)^2}$$ 2.$$\sum_{n=2,even}^{\infty}\frac{\exp(-an^2)}{(b-n^2)^2}$$ with $a$ and $b$ $\in ...
2
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3answers
59 views

Summation of a finite series

Let $$f(n) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+...+ \frac{1}{2^n-1} \forall \ n \ \epsilon \ \mathbb{N} $$ If it cannot be summed , are there any approximations to the series ?
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4answers
114 views

Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$

For $a,b,c>0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$ how can I find the limit of sequence above? Provide me a hint or full solution. thanks ^^
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43 views

Uniform convergence of $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$.

Given that the series $\sum_{n=2}^{\infty}\dfrac{\mathrm{ln}(n)^{p}}{n}$ is convergent iff $p<-1$. Show that $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$ is uniform convergent as $x \in ...
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0answers
26 views

Growth of partial sums of a divergent series

I am trying to find how the product $\prod_{n=0}^Me_n$ grows with $M$, when $$e_n=1+\frac{a_1}{t+n}+\frac{a_2}{(t+n)^2}+\dots$$ with large $t$. So as a first estimate I took $e_n=1+\frac{a_1}{t+n}$ so ...
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1answer
54 views

Compute limit of the sequence given by $x_1=1$, $x_{n+1}^2=\frac{x_n+3}{2}$

Let $\left(x_n\right)$ be a real sequence such that $x_1=1,x_{n+1}^2=\dfrac{x_n+3}{2},\forall n\geq 1$.Compute $\lim_{n\to+\infty} 3^n\sqrt{\dfrac{9}{4}-x_n^2}$
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1answer
63 views

3rd grade exercise: “make your own turning pattern”

My 8 year old has been given a worksheet of numeric sequences, e.g. "what are the next numbers in the sequence 11, 12, 14, 15, 17, ..." and "make your own number pattern" and "make your own colour ...
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1answer
120 views

Uniformly convergence?

I'm not sure wether or not the following sum uniformly converge on $\mathbb{R}$ : $$\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$$ Can someone help me with it? (I can't use Dirichlet' ...
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3answers
34 views

Arithmetic progression where the sum of the first $p$ terms is $q$, and the first $q$ terms is $p$

The sum of first $p$ terms of an arithmetic progression (A.P) is $q$, and the sum of the first $q$ terms of the same A.P. is $p$. Find the sum of the first $p+q$ terms of the A.P.
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How prove this $\sum_{n=1}^{\infty}\dfrac{\zeta_{2}}{n^4}=\zeta^2(3)-\dfrac{1}{3}\zeta(6)$

show that $$\sum_{n=1}^{\infty}\dfrac{\zeta_{2}}{n^4}=\zeta^2(3)-\dfrac{1}{3}\zeta(6)$$ where $$\zeta_{m}=\sum_{k=1}^{n}\dfrac{1}{k^m},\zeta(m)=\sum_{k=1}^{\infty}\dfrac{1}{k^m}$$ is true? because ...
2
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1answer
64 views

Bounded sequence in Hilbert space contains weak convergent subsequence

In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence. Is there any short proof? Thanks a lot.
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302 views

$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number

I need help to prove the following result. $\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
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62 views

Summation involving subfactorial function

Inspired by this post: Does the following series converge; if so, to what value does it converge? $$ \sum_{n = 2}^\infty \left|\frac{!n \cdot e}{n!} - 1\right|$$ I am looking for a closed form for ...
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1answer
43 views

Infinite Series Problem Using Residues [duplicate]

Show that $$\sum_{n=0}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{2a}\coth\pi a+\frac{1}{2a^2}, a>0$$ I know I must use summation theorem and I calculated the residue which is: ...
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1answer
34 views

How to derive the sum of an arithmetic sequence?

I'm attempting to derive a formula for the sum of all elements of an arithmetic series, given the first term, the limiting term (the number that no number in the sequence is higher than), and the ...
4
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1answer
109 views

Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$

By integral test, it is easy to see that $$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$ converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$] I am ...
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44 views

Prove the convergence of the sequence.

Prove the convergence of the following sequence: $$x_1 = \sqrt{a}$$ $$x_{n+1} = \sqrt{a + x_n}$$
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3answers
45 views

What is the sum of this infinite series? Which one is it, Taylors? Binomial?

I am trying to figure which formula to use for this one. $$\displaystyle\sum\limits_{x=-1}^{-\infty} -x(1-y)p(1-p)^{-x}+\sum\limits_{x=1}^\infty xyp(1-p)^x$$ where $0<y<1$, and $0<p<1$. ...
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3answers
56 views

Infinite Series question [duplicate]

The first one, the effective resistance is $2R$, then $5R/3$ then $13R/8$ etc.... My job is to find the pattern/equation so I can find the total resistance when $20$ resistors are connected. Of ...
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2answers
62 views

convergence of series with absolute value

prove or show false: if $\sum_{n=1}^{\infty}\left |a_{n} \right |$ converges, then $\sum_{n=1}^{\infty}\frac{n+1}{n}a_{n}$ converges as well. Thank you very much in advance, Yaron.
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0answers
28 views

analysis: limit of product of sequences [duplicate]

I would really appreciate help with this question: Show that if the limit $\lim_{n \to \infty} {a_n} = 0$ and sequence $\{b_n\}$ is bounded, then $\lim_{n\to \infty} a_nb_n =0$ thanks
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1answer
31 views

what does “in wide sense” mean?

I came across the statement "the sequence increases(in wide sense)". So my doubt is what does author mean by wide sense?I came across this in number theory book
4
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2answers
89 views

Why doesn't the Weierstrass approximation theorem imply that every continuous function can be written as a power series?

I hope that my question in the title is well formulated. I am a little bit confused with the next exercise from a book: Argue that there exists functions $f \in C[0, \frac{1}{2}]$ that cannot be ...
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1answer
29 views

series convergence

i ran into this question: prove or show false: if $\sum_{n=1}^{\infty}a_{n}$ is a converging series, but the series $\sum_{n=1}^{\infty}a_{n}^2$ diverges, then $\sum_{n=1}^{\infty}a_{n}$ is ...
7
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1answer
72 views

methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$

As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which ...

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