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

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1
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1answer
30 views

Prove or disprove: if the limit of $a_{n}$ goes to $\infty$, then $a_{n}$ has a minimum number.

Prove or disprove: if the limit of $a_{n}$ goes to $\infty$, then $a_{n}$ has a minimum number. Well I know from the definition of a series that goes to $\infty$ that there is a number $K>0$ that ...
2
votes
2answers
57 views

Prove or Disprove: If $a_n$ was positive and its limit is positive then the limit of the $\sqrt[n]{a_n}$ is $1$

If $a_n$ was positive and its limit is positive then the limit of the $\sqrt[n]{a_n}$ is $1$. I think it has to do with multiplication to start here,but I couldnt figure it out.
1
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0answers
31 views

a real analysis question,I need to prove whether two sequences are equidistributed or not,really need some help!

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
-3
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1answer
51 views

Use Sigma Formulas and Find the Value of the Sum: $\sum_{i=1}^{n}(7+6i)^2 $

I have $294n+252n^2+252n+2n^3+n^2+2n^2+\frac n 6$ The number seems too large.
0
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0answers
10 views

Approximating an Appell series with a second-order polynomial

Taking this Appell F1 hypergeometric series $$ f(t)=F_1\left(2;\frac{3}{2} (1-m),\frac{3}{2} (\lceil r\rceil +m-1);3;\frac{\frac{r}{2} t^2 \left(1-\frac{r}{\lfloor r\rfloor +m}\right)+\lfloor r\rfloor ...
2
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4answers
32 views

How can I find if $\sum_{n=2}^\infty {(-1)^n*4 \over (ln(n))^2} $ converges or diverges using the alternating series test?

$$4\sum_{n=2}^\infty {(-1)^n \over (\ln(n))^2} $$ If $4 \over (\ln(n))^2$ = $u_n$, then $u_n$> 0, and: $$\lim_{n\to\infty} u_n = 0 $$ But there is one more test to prove convergence which says ...
0
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1answer
51 views

How can I solve like this exercise

Let we have the following initial value problem : $$y'=f(x,y)=e^y$$ With the condition $y(0)=0$ Find the largest interval $|x| \le a $ makes the initial value problem has an unique solution ...
0
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0answers
21 views

limit of recurrence serie by derive

how to define limit this series $$U_{n+1} = {df(U_n)\over dU_n}$$ $f(x)$ function derivable in Class $C^n$, $f^{(n)}(x)$ function not null and $U_0 \gt 0$
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0answers
21 views

True/false statements about convergence of a sequence

Let $(a_n)$ be a sequence. True or false: If $\lim_{n\to\infty}(a_{2n}-a_n)=0$ then $(a_n)$ converges. If $(a_n)$ converges then $\lim_{n\to\infty}(a_{2n}-a_n)=0$ Intuitively, I ...
0
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1answer
53 views

How simplify this sum?

I need help to simplify this sum : $$\sum_{i=0}^{x-1}\left(1-\dfrac{1}{2^i}\right)^{m-1}$$ Is it possible to simplify it ? Thank you.
0
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0answers
39 views

When can we assume this identity about series?

Suppose we are given that $$\sum_{i =a} ^b f (i)g (i) =\sum_{i =a} ^b f (i)h (i) \tag {1} $$ when can we conclude that $$\forall i =a,…,b:g(i)=h(i) \tag {2} $$ ? For example, if $g, h $ are both ...
2
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1answer
49 views

How to evaluate $\sum\limits_{n=1}^\infty (-1)^{n-1} \ln (1+\frac1n)$

Can someone help me evaluate the sum of this series through elementary means? $$\sum_{n\geq 1}(-1)^{n-1} \ln \left(1+\frac1n\right)$$
2
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4answers
85 views

Can I find $\sum_{k=1}^n\frac{1}{k(k+1)}$ without using telescoping series method?

I am interested to find $$\sum_{k=1}^n\frac{1}{k(k+1)}$$ without using telescoping series method. I tried very hard but still could not think of a way to find it without using telescoping series ...
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4answers
49 views

Root test for $\sum_{n=8}^\infty \left(1+\frac{3}{n}\right)^{n^2}$

I got that it would be infinity and diverge, but my answer seems to be incorrect. What did I do wrong? $$\lim_{n\to\infty}\left(1+\frac3n\right)^{n^2}= \lim_{n\to\infty} ...
1
vote
1answer
50 views

evaluate trigonometric

To all the genius out there , here is a question about expresssing summation of hyperboilc functions : First of all, I've already proved that: $$\sinh(x + 1)- \sinh(x) = (-􀀀1 + \cosh (1)) \sinh(x) ...
0
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0answers
34 views

Calculating Infinite Sums [on hold]

How do you calculate sum of infinitely sequenced numbers in an arithmetic or geometric progression? Please help me out! Thanks in advance!
0
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2answers
33 views

Could someone help me clarify the steps for this solution?

Given $$\sum_{n=1}^\infty \frac{1}{n^6} = \frac{\pi^6}{945},$$ calcuate $$\sum_{n=1}^\infty\frac{1}{(n+2)^6}.$$ Solution: $$\sum_{n=1}^\infty\frac{1}{(n+2)^6} = \frac{1}{3^6} + \frac{1}{4^6} + ...
0
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0answers
28 views

Series verification

Anyone can tell me what series is this? As I heard that this kind of series already been long understood. I am required to calculate the value of $P_2$ from the 1st sequence, the value of $P_2$ is ...
4
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4answers
63 views

The sequence $\frac{2}{2-u_n}$ diverges

Let $(u_n)$ be a sequence defined with $u_{0}$ a real number such that $u_0 \notin \{0,1,2\}$ and $$u_{n+1} = \frac{2}{2-u_n}$$ Prove that $(u_n)$ diverges. I try to use the fact that this sequence ...
2
votes
1answer
34 views

convergent series and divergent series

Hi I have two questions. First, $\sum_{n=1}^\infty \frac{n}{n^3+1}$. Is it divergent or convergent? I think it seems like it is positive and decreasing function so we can apply integral test. ...
3
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2answers
39 views

Multiple part problem concerning the proof that $\sum_{k=1}^n k^3=\left(\frac{n(n+1)}{2}\right)^2$ by induction

So I'm having trouble with $c,d$ and $e$. For $c$ so far I have: Inductive Hypothesis: $(\frac{n(n+1)}{2})^2 = (\frac{(k+1)(k+2)}{2})^2$ is that correct?
3
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2answers
54 views

How can I find the convergence/divergence of $\sum_{n=1}^\infty {n! \over n^n}$

Using either root test or ratio test. I have the feeling that it is the root test, I'm not sure how to proceed from this: $$ \sqrt[n]{n! \over n^n}= {(n!)^{1\over n} \over n} $$
1
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1answer
22 views

Derive an exact formula (solve the recurrence definition) for the following recursive sequence:

Derive an exact formula (solve the recurrence definition) for the following recursive sequence: $s_n = 2_{s_n-1} - s_{n-2}$ where $n \ge 2$, and $s_0 = 4$, $s_1 = 1$. So I saw someone solving this by ...
1
vote
3answers
63 views

How can I find if $\sum_{n=1}^\infty {n! \over 10^n} $ converges or diverges?

$$\sum_{n=1}^\infty {n! \over 10^n} $$ I wasn't sure on which method to use, I think the ratio test might work, but I'm stuck. Here's what I have so far: $a_n$= $n! \over 10^n$ & ...
1
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1answer
35 views

Proof that $\lim \frac{a_n}{1+a_n^2} = 0 \implies \lim a_n = 0$

I´ve tried some exercises about sequences convergence, particularly: Let $a_{n}$ be a sequence such as $\displaystyle\lim_{n\rightarrow\infty}\frac{a_{n}}{1+a_{n}^2}=0.$ Prove that $a_{n}$ ...
1
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2answers
34 views

Prove that every natural number is a limit of a subsequence

I need to prove that every natural number is a partial limit (limit of a subsequence) to the sequence ${{a}_{n}}=n-{{\left\lfloor \sqrt{n} \right\rfloor }^{2}}$ . I already found that 0 is a partial ...
2
votes
2answers
46 views

How to find convergence/divergence of this series

$$\sum_{n=1}^\infty {1+\cos(n) \over n^2}$$ I used the comparison test and said that $\sum_{n-1}^\infty {1 \over n^2}$ is comparable and also larger than $\sum_{n=1}^\infty {1+\cos(n) \over n^2}$, ...
2
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1answer
74 views

Showing that $\sum\limits_{n=1}^{\infty} (a_1+2a_2+…+na_n)/n(n+1) = \sum\limits_{i=1}^n a_n $

Let $\sum\limits_{n=1}^{\infty} a_n$ a series of positive terms convergent. Show that $\sum\limits_{n=1}^{\infty} \frac{a_1+2a_2+...+na_n}{n(n+1)}$ converges to the same value of $\sum\limits_{i=1}^n ...
1
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0answers
25 views

How would I determine if this infinite series is convergent or divergent using the limit comparison test?

$$\sum_{n=1}^\infty {2^n \over3+4^n}$$ My thinking is that $4^n$ will grow much more rapidly than $2^n$, and the +3 in the denominator is negligable. Therfore, I should compare it to ...
5
votes
1answer
59 views

Solving $y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}$

There is the trivial $y=0$, but beyond that, could there be further solutions for $y$ in terms of $x$ such that $$y=\prod_{n=1}^{\infty}\frac{d^ny}{dx^n}\mbox{ pointwise}$$ ? I posed this problem ...
0
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1answer
28 views

estimates for Abel's theorem

Suppose $a_1,a_2,\dots$ is a sequence of real numbers with $\displaystyle\sum_{n=1}^\infty a_n =s<\infty$. For $0<z<1$, define $f(z):=\displaystyle\sum_{n=1}^\infty z^n a_n$. By Abel's ...
0
votes
1answer
56 views

If $(y_{2n}-y_n) \to 0$ then $\lim_{n\to \infty} y_n$ exists

Assume $$\lim_{n\to \infty} (y_{2n} - y_n)=0$$ then $$\lim_{n\to \infty} y_n$$ exists. I know it's not true, and I can see a sequence that disprove that $(1,1,2,1,3,2,4,1,5,3,...)$ but I want a ...
1
vote
1answer
52 views

Infinite series $\sum_{n=1}^{\infty}nx^{n+1}$ does not comply to any of my (known) tests

I am attempting to find the interval of convergence for $$\sum_{n=1}^{\infty}nx^{n+1}$$ The lower bound, x = -1, would be tested by determining if $$\sum_{n=1}^{\infty}n(-1)^{n+1}$$ diverges. ...
0
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0answers
27 views

Given $f=x^4+x+1 \in \mathbb Z_{2}[x]$ is primitive, write down an $m$-sequence ${a_n}$ associated to $f$

I'm not sure how to solve this question exactly. I know that the period will be 15 but I don't know how to construct the $m$-sequence.
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0answers
24 views

Geometric and arithmetic sequence [on hold]

The fourth term of a geometric sequence is -64.The product of the first and third terms is 16.Which of the following statements is/are true? A.The first term and common ratio have opposite signs B.The ...
4
votes
1answer
50 views

Find a series convergent $\sum a_n$ such that $\sum\sqrt{a_n/n}$ diverges.

This is part of an exercise 8.22 from Apostol's Mathematical Analysis. I've looked at things like, $a_n=1/\sqrt n-1/\sqrt{n+1}$, and $a_n=1/\log n^{\log n}$, but I can't seem to find anything that ...
2
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1answer
19 views

Finding a sequence of step functions converging uniformly to $f(x)=\sum_{n=0}^\infty 2^{-n}\mathbb{1}(x>q^n)$

As I was revising my Real Analysis course, I came across this strange problem on series of functions. If anybody can verify that the first and last parts of the proof are correct and give me hints ...
1
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1answer
34 views

Is $H(\theta) = \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos (2\pi n_k \theta)$ for a given sequence $n_k$ equal a.e. to a continuous function?

I am studying Furstenberg's article Strict ergodicty and transformation of the torus and I'm stuck with the following construction. Define sequence $(v_k)_{k \in \mathbb{N}}$ as $v_1 =1, ...
1
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4answers
35 views

Proving that $\lim_{n\rightarrow\infty}n(a_{n+1}-a_{n})=1 \implies a_n$ diverges to $\infty$

I'm trying to prove that given a sequence $a_{n}$ such as $\displaystyle\lim_{n\rightarrow\infty}n(a_{n+1}-a_{n})=1,$ then $a_{n}$ diverges to $\infty.$ I'm lost searching a path to prove it. I ...
10
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3answers
115 views

Proving the limit of a nested sequence

I have trouble proving the next sequence limit: $\displaystyle\lim_{n\rightarrow\infty}(x_{n}-\sqrt{n})=\frac{1}{2}$ where $x_{n}=\sqrt{n+\sqrt{n-1 ...\sqrt{2+\sqrt{1}}}}.$ I've had a lot of ...
2
votes
2answers
50 views

Radius of convergence of the series $ \sum\limits_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{3\cdot6\cdot9\cdots(3n)}x^n$

How do I find the radius of convergence for this series: $$ \sum_{n=1}^\infty (-1)^n\dfrac{1\cdot3\cdot5\cdot\cdot\cdot(2n-1)}{3\cdot6\cdot9\cdot\cdot\cdot(3n)}x^n $$ Treating it as an alternating ...
0
votes
2answers
33 views

What formula can I use to identify numbers in the pattern 2 7 10 15 18 23 [on hold]

I have an algorithm that loops from 1 to n and need to pick out those numbers. E.g. to find multiples of 3: for n in [0..100] if n % 3 == 0 //do something
1
vote
1answer
39 views

Infinite summation of a trigonometric series

$\sum_{n=1}^{n=\infty}\sin(\frac{n\pi x}{L})\sin(\frac{n\pi y}{L})\surd(k^2+\frac{n^2 \pi^2}{L^2})$ I am trying to solve the above summation. I still could not figure out if this summation converges ...
2
votes
3answers
30 views

Proving convergence of sequences

I've tried to prove the following proposition but my attemps have failed: Given sequences $a_{n}$ and $b_{n}$ such as $\displaystyle\lim_{n\rightarrow\infty}(a_{n}^{2}+b_{n}^{2}+a_{n}b_{n})=0$ then ...
2
votes
0answers
28 views

Determine whether this series converges (proof verification)

Determine whether the following series converges: $$\sum_{n=2}^\infty\frac{(-1)^n}{\sqrt{n}-(-1)^{[\sqrt{n}]}}$$ where $$[x]=\max\{k\in\mathbb{Z}: k\leq x\}$$ My attempt: First I write ...
0
votes
2answers
34 views

is there any mathematical model how the guitar strings are related?

I'm just curious to know the mathematical relationship between guitar strings and how their frequency changes with the variation of guitar's string length and thickness. Say, I'm vibrating some node ...
2
votes
3answers
410 views

Product of two infinite sequences

Let $p_i$ be reals in (0,1) such that $\sum_1^{\infty} p_i=\infty$ and $\sum_1^{\infty} (1-p_i)=\infty$. Prove that $\sum_1^{\infty} p_i(1-p_i)=\infty$. I know a probabilistic proof (follows from ...
2
votes
2answers
21 views

Sufficient conditions to guarantee Cesaro summarble

Is is true that every nonnegative, bounded series in $R$ is Cesaro summarble? Is there a list of sufficient conditions on series to guarantee Cesaro summarble?
0
votes
2answers
28 views

Can we write $\sum_{k=1}^\infty a_k \frac{\sin k}{k}\leq a_\nu \sum_{k=1}^\infty \frac{\sin k}{k}$?

We known that $$\sum_{k=1}^\infty \frac{\sin k}{k}$$ converges. If $a_k\in\mathbb R$ and $a_k\leq a_\nu, \ \forall k\in\mathbb N,$ can we infer that $$\sum_{k=1}^\infty a_k \frac{\sin k}{k}\leq a_\nu ...
3
votes
0answers
46 views

Proving a strictly decreasing sequence which tends to zero is positive

Suppose $(a_n)$ is a strictly decreasing sequence such that $a_n\underset{n\to\infty}{\rightarrow}0$. I'm asked to prove that $(a_n)$ is positive. My approach: suppose there is a negative element ...