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

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14 views

little oh notation problems with exercise.

I am having problems with the use of the little oh notation my professor is adopting in the solutions to some exercises. As an example I do not understand why $$ \frac{1}{n}\ln(a+o(1)) = ...
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1answer
36 views

Convergence of sequences such as $ B(n)=1/\sqrt{n^2+1}+\dots+ n/\sqrt{n^2+n}$

Examine the following arithmetic sequences if they converge or do not.The first one is $$ B(n)=1/\sqrt{n^2+1}+\dots+ n/\sqrt{n^2+n}$$ and the second $$C(n)=n/(n^2+1)+\dots+n/(n^2+n)$$ It was on our ...
3
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1answer
53 views

$\lim (x_n-y_n)=0$ $\implies$ $\lim \Big(\dfrac {x_n}{y_n}\Big)=1$ ?

If $(x_n)$ , $(y_n)$ are sequences of non-zero real numbers such that $\lim (x_n-y_n)=0$ , then is it true that $\lim \Big(\dfrac {x_n}{y_n}\Big)=1$ ?
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4answers
42 views

How do I prove convergence of this recursive sequence, what's the limit?

I have this sequence: $c_n = c_{n-1} + \frac{0.01}{n}, \ \ c_1 = 0.01$ How do I prove the convergence of this, and what is the limit?
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3answers
60 views

Convergence of the series $\sum n!/(n^2+3)$

How can we test if this series diverges/converges? $$\sum_{n=1}^\infty\frac{n!}{n^2+3}$$ I tried D'Alembert's principle and tried to do $\frac{a_{n+1}}{a_n}$ but I'm stuck. Any help?
2
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0answers
32 views

Evaluation of a dilogarithmic integral

Problem. Prove that the following dilogarithmic integral has the indicated value: $$\int_{0}^{1}\mathrm{d}x ...
4
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4answers
80 views

Evaluation of the series $S(\omega)=\sum\limits_{k=0}^\infty (-1)^k {\alpha \choose k}\cos(k\omega)$

I had a problem evaluating the series \begin{equation} S(\omega)=\sum_{k=0}^\infty (-1)^k {\alpha \choose k}\cos(k\omega),\quad 0<\alpha<2,\quad \omega\in(-\pi,\pi) \end{equation} where ...
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0answers
21 views

Proving that an oscillatory recursive sequence converges

I have a recursive sequence that is quite similar to the one shown here Prove that a given recursion sequence converges with more complex definition. I know that my sequence is also bounded, but on ...
11
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0answers
77 views

Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?

Let $a_0=1,a_n=\tan{a_{n-1}}$. Then is $\{a_n\}_{n=0}^\infty$ dense in $\Bbb{R}$? I've drawn a map of this dynamical system and it seems that the sequence is dense on $\Bbb{R}$.
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2answers
47 views

Two questions about convergence in measure

I am currently studying for my analysis comprehensive exam and have a few questions about convergence in measure. First of all I know that for a sequence $\{f_n\} \in L^p(E)$, $1 \le p < \infty$, ...
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0answers
15 views

Induction On Frequency Properties Of Sequences

The title of the question may be a bit misleading, sorry, but I was at a loss of what else to call it. I have two sequences of integers which are generally quite random, except that the rate at ...
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1answer
24 views

Number of terms in an Arithmetic Series between 2 limits

consider an Arithmetic Series such as 2 + 5 + 8 + 11 + 14 + ... (a = 2, d = 3) Also, consider an Upper Bound, say, U = 13 I'm hoping to get an explicit expression for the NUMBER of TERMS, T, ...
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0answers
15 views

Decomposing non-periodic function into exponential series

Is it possible to represent a non-periodic function as a series of exponents, and if yes, how?
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1answer
25 views

Convergence of Bounded Sequence Satisfying $2c_n \leq c_{n+1} + c_{n-1}$

Let $(c_n)$ be a bounded sequence satisfying $2c_n \leq c_{n+1} + c_{n-1}$. (a) Let $x_n = c_{n+1}-c_n$. Show that $(x_n)$ is increasing. (b) Show that $x_n \to 0$ as $n \to \infty$. ...
3
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1answer
19 views

Convergence of $\sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^2\right)^{-\alpha}$

For which $\alpha$ (depending on $n$) does $$\sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^2\right)^{-\alpha}$$ converge? Examples: For $n=1$ the series turns out to be $$\sum_{i=1}^\infty ...
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1answer
30 views

Analysis Arithmetic series .Verify which of the following sequences converge.

Verify which of the following sequences converge.$$1.A(n)=\sum_{n=1,n=+00}(1/(n^{1+1/n})$$ $$2. B(n)=(1/\sqrt{n^2+1})+.......n/\sqrt{n^2+n}$$ $$3.C(n)=(n+cos(n^2))/(n+sin(n)) $$ .For the 3th one ...
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1answer
36 views

How can I find the area of the shadow? 3 [on hold]

how can i find the surface area of circular coin placed 's' distance from the source(light producing source) which produces a shadow 'd' distance from the surface.the circular coin is placed between ...
3
votes
2answers
85 views

Evaluation of $\sum n a^n$ using telescoping property

Show that the series $$\sum\limits_{n=1}^{\infty}n a^n = \frac{a}{(a-1)^2} $$ for $|a|<1$ using the telescoping property. I know how to do this using other methods. But the exercise ...
3
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1answer
81 views

Calculate $\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$

Let $\alpha$ be a positive number. Calculate $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$ Edit: I have deleted my attempt, it didn't seem to lead me anywhere and I discovered ...
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2answers
47 views

Compute $0.y_1y_2y_1y_2y_1y_2…$

Any tips? I'm taking $$Sn = \frac{y_1}{10} + \frac{y_2}{100} +\frac{y_1}{1000} + $$ and then I find $$\frac{S_n}{100}=...$$ I do $$Sn - \frac{S_n}{100}$$ and go on from there but I can't find the ...
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2answers
60 views

How to find the limit of $ {n! e^n}/{n^{n+1/2}} $?

What is the value of this limit and how to find it? $$ \lim_{n \to \infty} \frac{n! e^n}{n^{n+\frac{1}{2}}} $$ Can we use L'Hospital rule here? I tried but failed that how to do it.
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0answers
111 views

What is the value of the power series with rising factorial coefficients?

The power series given by \begin{equation} S(\alpha,n,x) = \sum_{k=0}^\infty a_k(\alpha,n)x^k,\quad n\in \pmb{N}_0,\quad x\in\pmb{R} \end{equation} where \begin{equation} a_k(\alpha,n)=\left[ \alpha ...
1
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0answers
19 views

Powerful numbers in Pell solutions (or, more generally, any Lucas sequence)

There are several definitive results regarding perfect powers in the Pell numbers — e.g., the only perfect power is $P_7=169=13^2$. On the other hand, when it comes to powerful numbers, I've only ...
5
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2answers
72 views

Binomial Expansion, Taylor Series, and Power Series Connection

1) Is there a reason why the binomial expansion of $(a+x)^n$ is the same as a Taylor series approximation of $(a+x)^n$ centered at zero? 2) The binomial expansion of $(a+x)^n$ is $a^n + na^{n-1}x ...
0
votes
1answer
74 views

Convergence of the series $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ [on hold]

I need to find out whether this series converges or diverges. $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$ Can someone help how to solve it?
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2answers
31 views

Convergence test of certain series

I need to find out whether this sequence converges or diverges using limit comparison test. $\sum_{n=2}^{\infty}\frac{\sqrt{n+2}-\sqrt{n-2}}{\sqrt n}$ I've tried it with the use of sequence ...
3
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2answers
40 views

Differentiability for the uniform limit of a uniformly bounded sequence of functions

Let a sequence $\{f_n\}\subset C^1(\mathbb{R})$ and $f\in C(\mathbb R)$ such that $f_n \to f$ uniformly and $f_n, f'_n$ are uniformly bounded. Question : is $f \in C^1(\mathbb R)$ ?
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2answers
86 views

How to solve this and what is this number called? [duplicate]

What is the real number called to which the sequence $$\gamma_n =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} - \log _e n$$ converges and what is the radius of convergence?
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1answer
30 views

Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely

I'm trying to solve the following Problem: Let $(X_n)_{n\ge 1}$ be a sequence of real valued random variables defined on some probability space $(\Omega, \mathcal{A},P)$. Assume that there ...
0
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1answer
25 views

Limit of a function series

Compute: $ \lim_{ x \rightarrow 1^{-}} \sum_{n=1}^\infty \dfrac{(-1)^{n-1}x^n}{n(1+x^n)} $ I compute: $$ \sum_{n=1}^\infty \lim_{ x \rightarrow 1^{-}}\dfrac{(-1)^{n-1}x^n}{n(1+x^n)}\\ = ...
3
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1answer
39 views

If a series has the same sum under any rearrangement, then is it absolutely convergent?

Let $(V,\| \cdot \|)$ be a Banach space. Let $\{a_n\}$ be a sequence in $V$ such that $\sum a_n$ converges. Assume that for every bijection $f:\mathbb{N}\rightarrow \mathbb{N}, \sum a_n = \sum ...
3
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2answers
54 views

What is the value of the limit $\lim_{a\searrow 0}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{a}}$?

Clearly the series $$ \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{a}} $$ converges (conditionally), as an alternating series of as absolutely decreasing sequence, for all $a>0$. The question is: What ...
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0answers
89 views

Closed formula for the numbers of the form $\sqrt{1+\sqrt{4+\sqrt{9}}}$

how can i find the formula for the nth term of this series? SQ = square root $\sqrt{1}$ = 1 $\sqrt{1 +\sqrt{4}}$= sq rt of of 3 $\sqrt{1 +\sqrt{4+\sqrt{9}}}$=1.909385061 $\sqrt{1 ...
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2answers
40 views

To test the convergence of series 2

To test the convergence of series 2 $\displaystyle \frac{a+x}{1!}+\frac{(a+2x)^2}{2!}\frac{(a+3x)^3}{3!}+...\infty$ My Attempt: $$\begin{align} \frac{u_n}{u_{n+1}} & = ...
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2answers
125 views

Sum of $1+\frac{1}{2}+\frac{1\cdot2}{2\cdot5}+\frac{1\cdot 2\cdot 3}{2\cdot 5\cdot 8}+\cdots$

I am trying to find out the sum of this $$1+\frac{1}{2}+\frac{1\cdot2}{2\cdot5}+\frac{1\cdot 2\cdot 3}{2\cdot 5\cdot 8}+\frac{1\cdot 2\cdot 3\cdot 4}{2\cdot 5\cdot 8\cdot 11}+\cdots$$. I tried with ...
0
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2answers
26 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
2
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3answers
63 views

Why is $ A_1 x + … + A_n x^n $ a solution of $ \sum_0^{n} (-1)^n \frac{x^n}{n!} \frac{d^n y}{d x^n} = 0 $?

I was playing(/fiddling) around with some maths and I saw this pattern( where $ A_n $ is a constant.): $ A_1 x $ is a soultion of: $$ \frac{y}{x} - \frac{dy}{dx} = 0 $$ $ A_1 x + A_2 x^2 $ is a ...
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1answer
60 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
1
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2answers
56 views

When can an infinite sum and complex integral be interchanged?

Are there some conditions under which the following two are equal? $$\displaystyle \oint_C \sum f_n(z)= \sum \oint_C f_n(z)$$ In the case of real valued functions, the condition $f_n(z) \geq 0$ ...
4
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1answer
41 views

Find $\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right)$

Find $$\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right).$$ I have tried rewriting the sum in a clever way, applying the Mean Value Theorem or Stolz-Cesaro ...
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1answer
33 views

Sequence problem with common difference

I have a problem with one sequence word problem. You add $1000$ to your bank account and withdraw $62$ the first year, and withdraw $4$ more every year than the year before. I have to find how much ...
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0answers
14 views

Convergence of Series of Complex Numbers with Decreasing Modulo (non-zero imaginary part)

Let $(a_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}$, $\theta$ not a multiple of $2\pi$, the series $\sum_{n\geq1} ...
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1answer
30 views

Question about prove of Absolute Convergence Test?

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I have a question about the proof of the Absolute Convergence Test (theorem 2.7.6 on page 65). The author states that ...
3
votes
2answers
66 views

Limit of a sequence of averages (three variables)

Let $a_0 = 0$, $a_1 = 0$, $a_2=1$ and for $n>2$, $a_n = \dfrac{a_{n-1}+a_{n-2}+a_{n-3}}{3}$. Consider $\lim\limits_{n \to +\infty} a_n$. Using a python script I found that $a_n$ tends to ...
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1answer
40 views

looking at the alphabet ,the letters are numbered 1-26 ,

looking at the alphabet ,the letters are numbered 1-26 , such that 1 =one=15+14+5=34 (O=15, N=14, E =5 ) 2=two=20+23+15=58 (T=20, W=23, 0=15) 3=three =56 4=four=60 ...
0
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1answer
37 views

To test the convergence of series 1

To test the convergence of the following series: $\displaystyle \frac{2}{3\cdot4}+\frac{2\cdot4}{3\cdot5\cdot6}+\frac{2\cdot4\cdot6}{3\cdot5\cdot7\cdot8}+...\infty $ $\displaystyle 1+ ...
7
votes
4answers
332 views

Test the convergence of a series

To test the convergence of a series: $$ \sum\left[\sqrt[3]{n^3+1}-n\right] $$ My attempt: Take out $n$ in common: $\displaystyle\sum\left[n\left(\sqrt[3]{1+\frac{1}{n^3}}-1\right)\right]$. So this ...
0
votes
1answer
31 views

Relation between limit of functions and sequences

I need to prove that the sequence $$ \frac{\sqrt{n}}{\log n}$$ diverges. I know that $$\lim \frac{\sqrt{x}}{\log x} = \infty$$. Is there any theorem that relations the limit of the function with the ...
36
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2answers
1k views

Geometry problem involving infinite number of circles

What is the sum of the areas of the grey circles? I have not made any progress so far.
2
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0answers
74 views

How prove this sequence $u_{m}=v_{m}$

Question: Assume that $m$ is a positive integer, define the sequence $$\{u_{k}\},\{v_{k}\},u_{0}=v_{0}=u_{1}=v_{1}=1$$ and for any real number $a_{i},i=\{1,2,\cdots,m-1\}$, $$\begin{cases} ...