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

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3answers
1k views

Summing Finite terms of Harmonic series: $\sum_{k=a}^{b} \frac{1}{k}$

How do I calculate sum of a finite harmonic series of the form : $$\sum_{k=a}^{b} \frac{1}{k} = \frac{1}{a} + \frac{1}{a+1} + \frac{1}{a+2} +\cdots \frac{1}{b}.$$ Is there a general formula for ...
5
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4answers
154 views

Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?

How can we evaluate this series $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$ Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how ...
3
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1answer
68 views

distribution of $\cos(\omega_0 n)$ where n are integers?

Assume we have the sequence $\,x[n]=\cos(\omega_0 n),$ where $n$ are integers. If we suppose these are realizations of a random variable, what would be the p.d.f. of that random variable?
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1answer
85 views

$\lim_{n \to \infty} s_2(n) = n + {n-1\over2^2} + {n-3\over 3^2} + \ldots + {1 \over n^2}$ : surely infinite, but related to something?

I have the series $ \small b = 1 + (1+ \frac 14) + (1 + \frac 14 + \frac 19) + \ldots $, where the partial sums are $$ s_2(n) = n + {n-1\over2^2} + {n-2\over 3^2} + \ldots + {1 \over n^2}$$ This ...
9
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1answer
312 views

Closed form for $a_{n+1} = (a_n)^2+\frac{1}{4}$

I've been given the following sequence: \begin{align*} &a_0 = 0; \\ &a_{n+1} = (a_n)^2+\frac{1}{4}. \end{align*} I also have to prove that whatever I come up with is correct, but that will ...
-2
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1answer
143 views

new to matlab please help me..

i don't know what site my question will fit but i saw a matlab tag so please consider my question.. today, we have an activity and no one finished it. so my professor decided to make the activity as ...
1
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2answers
54 views

A sequence raised to a sequence

I was wondering if the following is true... If $(a_{n})$ is a sequence of positive terms converging to $a$ and $(b_{n})$ is a real sequence converging to $b$, then the sequence $(a_{n}^{b_{n}})$ ...
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2answers
31 views

Solving for $n$ in a geomtric progression

Given the general term of geometric sequence: $a_n = \dfrac{x}{2^n}$ I would like to solve for the value of n that makes $a_n =1$. My work so far: \begin{align*} a_n &= \frac{x}{2^n}\\ 2^n &= ...
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0answers
40 views

Difference equation of series

Lets first get the preliminaries out of the way. Define $\Delta_+f(x)=f(x+1)-f(x)$. Now define $$C_j(x;a)=\sum_{m=0}^j(-1)^m\begin{pmatrix}j\\m\\\end{pmatrix}a^{-m}(x-m+1)_m$$ This can be rewritten ...
6
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2answers
139 views

Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$

I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some ...
1
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1answer
105 views

How to reduce this series / sumation?

I was solving a probability problem and I ended up with the solution of the form : $$ ...
1
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1answer
68 views

For which a this series converge? $\sum\limits_{n=1}n^{a+1}\int_n^{n+\log(n)}\frac{\arctan(t^a)}{1+t^{2a}}$

For which $a$ $\in$ $\mathbb{R}$ this series converge? $$\sum_{n=1}^\infty{n^{a+1}\int_n^{n+\log(n)}\frac{\arctan(t^a)}{1+t^{2a}}}$$
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0answers
58 views

How would you find a closed form expression for the following:

$$a_n = \frac{n}{n+1}a_{n-1} + \frac{n}{n-2}a_{n-2}\;,\;\;n\ge 3\; $$ For: $ 0 \leq n \leq 2 $, $a_n= n $
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1answer
75 views

Conditions for series convergence

In what conditions the series converges? $$\sum_{n=1}^{\infty}(2-e^a)(2-e^{a/2})\cdots(2-e^{a/n}), \quad a>0$$
2
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1answer
73 views

Some statement about Cauchy product of sequences

Assume that we have two sequences $(a_n)_{n \in \mathbb Z}, (b_n)_{n \in \mathbb Z}$ such that for each $l\in \mathbb N $ the sequence $\left(|n|^l a_n\right)_{n \in \mathbb Z}$ is bounded, there ...
2
votes
1answer
41 views

Series involving Marcum Q function

I would like to have a better form of this series: $$\sum_{k=0}^{\infty}\,\frac{1}{k!}\,\left(\frac{ab\sin(c)}{\sqrt{2}}\right)^{2k}\,Q_{k+\frac{3}{2}}\left(ab\cos(c),bx\right)$$ where ...
2
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1answer
126 views

Prove that $\limsup_{n\to\infty} |X_n|/n \le1 $ almost surely

Suppose {$X_n$} a sequence of random variables. If $\sum_{n=1}^{\infty}P(|X_n|>n)<{\infty}$ Prove that $$\limsup_{n\to\infty}\frac{ |X_n|}{n} \le1 $$ almost surely What i have done so far: I ...
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1answer
69 views

Help for solving this sequence

I couldn't solve the following sequence and couldn't even see any pattern:
6
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2answers
116 views

Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges

Let $f$ be a non-negative decreasing function on $[1,+\infty)$. Prove that the series $$\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$$ converges.
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1answer
95 views

Calculating radius for $\sum\frac{\sin{n^3x}}{n^2}$

Let $\displaystyle \sum\limits_{n=1}^{\infty}\frac{\sin{n^3x}}{n^2}$ be a series: a. Find where the series converge pointwise and where uniformally. b. Does its derivative is continuous? About A: ...
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2answers
458 views

Confused with proof that all Cauchy sequences of real numbers converge.

First the textbook proves that all Cauchy sequences are bounded, and so have a convergent subsequence, $\{a_{n_{k}}\}$ that converges to a limit, say $L$. Now we use this to prove that all Cauchy ...
1
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1answer
41 views

Proving $u_n(x)\le v_n(x), \sum\limits_{n=1}^{\infty}v_n(x)\to f$ uniformlly $\Longrightarrow\sum\limits_{n=1}^{\infty}u_n(x)\to g$ uniformly

Prove if $u_n(x)\le v_n(x)$ and $\sum\limits_{n=1}^{\infty}v_n(x)$ converges uniformly then also $\sum\limits_{n=1}^{\infty}u_n(x)$ converge uniformly I thought solving it by using Weierstrass ...
1
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1answer
98 views

A sequence to infinity

Does the infinite sequence have a possible closed form? Minus sign is not a mistake, it's in the right place. $$\sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\cdots}}}}}$$ The sequence of signs has period ...
5
votes
3answers
186 views

Computing the value of logarithmic series: $Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $

Given a series of the type: $$Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $$ How does one evaluate it? Something I noticed was: $$Q(1,n) = \ln(1) + \ln(2) + \ln(3)+ \cdots+\ln(n) = ...
0
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2answers
364 views

Formula for sequences

Can you guess a general generating rule for these 7 sequences ? 2 3 4 2 3 4 3 4 4 2 3 4 3 5 4 5 4 5 6 2 3 4 3 5 4 6 5 4 6 5 6 5 6 6 2 3 4 3 5 6 4 7 5 4 6 5 7 6 5 7 6 7 6 7 7 2 3 4 3 5 6 4 7 5 8 ...
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2answers
45 views

Series Question $\sum_{n=1}^{\infty}\frac{4^n}{7^{n+1}}$

I`m trying to check if the following series are convergent. $$1)\sum_{n=1}^{\infty}\frac{4^n}{7^{n+1}}$$ $$2)\sum_{n=1}^{\infty}(-1)^n\frac{5^n}{4^{n+2}}$$ what I did so far for the first one is: ...
0
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1answer
41 views

Proving convergence of a series

I need to show that the series of general term $$\tanh \frac{1}{n}+ \ln \frac{n^2-n}{n^2+1}$$ converges. I was thinking to use an equivalence as $n \rightarrow \infty$ We know that $ \tanh ...
1
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1answer
292 views

Doubt in the proof that $l^{p}$ is complete

I was looking at the proof that $l^{p}$ is complete with respect to the standard metric. Suppose $x^{(n)}$ is a Cauchy sequence in $l^{p}$. Then Given $\epsilon > 0$, $\exists\,\, n_{0} \in ...
3
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4answers
359 views

Proving that the limit $ \lim\limits_{n\rightarrow \infty} (n!)^{\frac{1}{n}}$ diverges to infinity [duplicate]

I came across this limit in some context: $$ \lim\limits_{n\rightarrow \infty} (n!)^{\frac{1}{n}}$$ I could only say that $n! > n$ implies the limit is greater than or equal to $1$. However, the ...
3
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2answers
308 views

Linked Arithmetic progression and Harmonic progression

I would like to give some introduction about the origin of my doubt and then put forth my doubt , so that people who attempt answering will know the context . ...
0
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1answer
88 views

Applying Stirling's formula in testing for convergence of a sum

I trying to figure which $\beta \in \mathbb{R}$ make the series $\sum_{n=1}^{\infty}\frac{(\beta n)^n}{n!}$ converge. I have tried two tests: ratio test, and approximation by Stirling's formula. I ...
20
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4answers
919 views

How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?

I would like to investigate the convergence of $$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$ Or more precisely, let $$\begin{align} a_1 & = \sqrt 1\\ a_2 & = \sqrt{1+\sqrt2}\\ a_3 ...
6
votes
1answer
128 views

How find $a_{n}$ if $a_{n+1}=\sqrt{2a_n+1}$

let $a_{1}=\dfrac{\sqrt{2}}{4}$ and such $$a_{n+1}=\sqrt{2a_{n}+1}$$ find $a_{n}$ my idea:let $a_{n}=\dfrac{1}{2}\cos{x_{n}}$ $$\Longrightarrow ...
3
votes
2answers
96 views

What is the pattern in this series: $\frac{r}{2} + \frac{4r^2}{9} + \frac{9r^3}{28} + \frac{16r^4}{65} + \dotsb$

A book gave me the following series, and asked for which $r\in \mathbb{R}$ does it converge: $$\frac{r}{2} + \frac{4r^2}{9} + \frac{9r^3}{28} + \frac{16r^4}{65} + \dotsb$$ I feel dumb because I ...
1
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2answers
123 views

A strengthening of Raabe's test: $\sum a_n$ diverges if $\frac{a_{n+1}}{a_n} \geq 1 - \frac{1}{n} - \frac{A}{n^2}$ for $A>0$

The usual form of Raabe's test says that if $a_n>0$ and if for large $n$, $\frac{a_{n+1}}{a_n}\leq 1-\frac{p}{n}$ for $p>1$, then $\sum a_n < \infty$. A proof I've seen of this relies on the ...
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5answers
382 views

Why is an infinite series not considered an infinite sum of terms?

According to, for example, this excellent page on beginner calculus, an infinite series is NOT an infinite sum of terms.* I'm not even sure what that's asserting. I think an infinite series is an ...
6
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2answers
104 views

Sequence of natural numbers

Numbers $1,2,...,n$ are written in sequence. It's allowed to exchange any two elements. Is it possible to return to the starting position after an odd number of movements? I know that is necessarily ...
1
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1answer
62 views

if $f_n\to f$ uniformly in $[a,b]$ then $f\in BV$

While preparing to test in calculus I found the question above: Let $f_n(x)\to f$ uniformlly on $[a,b]$. Prove or give counterexample: if $\forall n\in\mathbb N f_n(x)\in BV$ and $\exists M>0$ ...
0
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0answers
34 views

Caracterize this sequence of sequences

Characterize this sequence of sequences $ (u_{np}) : \displaystyle{\sum_{i=1}^{n} u_{np_i} \neq \sum_{i=1}^n u_{nq_i}}, \{p_i\}_{i=1}^n \neq \{q_i\}_{i=1}^{n}\ $
0
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2answers
305 views

Check with d'Alembert's ratio test the series $\sum_{n=1}^{\infty} \frac{1. 3. 5\dots(2n-1)}{3^n}$

I want to check with delambre test the following series: $$\sum_{n=1}^{\infty} \frac{1. 3. 5\dots(2n-1)}{3^n}.$$ So the first step I did is: $$\sum_{n=1}^{\infty}\frac{2(n+1)-1)}{3^{n+1}} ...
3
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3answers
370 views

interval of convergence of $\sum n \exp (-x \sqrt n)$

$$\sum^{\infty}_{n=1} n \exp (-x \sqrt n)$$ How to find the interval of convergence? Obviously, 0 is not in the interval because the series becomes divergent. could you help me?
4
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2answers
123 views

If $\sum a_k^2 /k$ converges, then $1/N \sum_1^{N}a_k \to 0$

I want to show that if $\sum a_k^2 / k$ converges, then $1/N \sum_{1}^Na_k \to 0$. Now, if $a_n \to 0$, then the result follows. But of course $a_n\to 0$ is not a necessary condition for $\sum ...
3
votes
1answer
91 views

For non-negative $f$ such that $\int_1^\infty |f'(t)|dt < \infty$, $\sum f(k)$ and $\int_1^\infty f(t)dt$ converge or diverge together

Suppose that $f\in C^1([1, \infty))$, $f>0$, and $\int_{1}^\infty |f'(t)|dt < \infty$. I want to show that $\sum_1^\infty f(k)$ and $\int_1^\infty f(t)dt$ are either both convergent or both ...
0
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1answer
187 views

Mix GP and AP question

I appreciate your good help: Two consecutive terms of a GP are the 2nd, 4th and 7th terms of an AP respectively. Find the common ratio of the GP.
2
votes
2answers
78 views

Absolute sum of Fourier coefficients

Let $f$ be a smooth and rapidly decaying function. Is it true that $\sum_n|\hat f(n)|\leq C \sum_n |f(n)|$ for some constant independent of $f$? Thanks! Can we say that $\sum_n|\hat f(n)|\leq C ...
1
vote
1answer
77 views

convergent $f(a_n)$ for divergent $a_n$

Could you give an example of a divergent $a_n$ such that: 1) $\exp (a_n)$ is convergent 2) $a^2_n - a_n + 1$ is convergent $a_n$ need not to be the same for two cases.
4
votes
4answers
277 views

Limits and exponents and e exponent form

So I know that $\underset{n\rightarrow \infty}{\text{lim}}\left(1+\frac {1}{n}\right)^n=e$ and that we're not allowed to see it as $1^\infty$ because that'd be incorrect. Why is then that we can do ...
2
votes
1answer
190 views

uniform convergence of $\sum\limits_{n=1}^{\infty}3^n\sin\left(\frac 1 {4^nx}\right)$ in $[1,\infty)$

I want to check the uniform convergence of $\sum\limits_{n=1}^{\infty}3^n\sin\left(\frac 1 {4^nx}\right)$ in $[1,\infty)$. The hint in the book was using Cauchy condition for uniform convergence. For ...
4
votes
2answers
99 views

Find for which values of $\alpha\in\mathbb R, f_n(x)$ converge uniformally

The question is "Find for which values of $\alpha\in\mathbb R$ such that $ f_n(x)$ converge uniformally in $[0,\infty)$ where $f_n(x)=n^\alpha x \dot e^{-nx} $". For $\alpha<0$, the ...
0
votes
1answer
73 views

Geometric progression problem

Please help me solve this: A man decided to save 50 cents on the first day and on each successive day double the amount saved the previous day. how long does it take for his total savings to be 1 USD ...