For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0answers
7 views

Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum

The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation $$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$ Alternatively they are defined inductively: $$U_0(x)=1 ...
0
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1answer
22 views

Finding the maximum value of a divergent series

I came across this divergent sum- $$\sum_{n=1}^\infty\frac{1}{n+1}$$ Now,a divergent sum does not a limit.So is it possible to get a maximum value for the sum or more specifically prove that ...
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1answer
18 views

how to write as geometric series $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ [on hold]

How would I write $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ as a sum of geometric series?
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1answer
6 views

Given common terms (and their position) between an arithmetic and geometric sequences, find the common ratio. [on hold]

The fourth, seventh and sixteenth terms of an arithmetic sequence also form consecutive terms of a geometric sequence. Find the common ratio of the geometric sequence
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1answer
23 views

Odd and Even Fourier Series Extension of $f(x)=x$ on $[0,\pi]$

I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only ...
1
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2answers
19 views

Forming a sequence from a Cauchy Sequence

Let $(a_{n})$ be a Cauchy sequence. Is $c_{n} = (-1)^{n}a_{n}$ also a Cauchy sequence?
5
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1answer
65 views

If $\sum a_n$and$\sum b_n$diverge, can$\sum \min\{a_n,b_n\}$converge? [duplicate]

Do there exist sequences $\{a_n\}$ and $\{b_n\}$ satisfying all of the following properties? $a_n>0$ and $b_n>0$ $\{a_n\}$ and $\{b_n\}$ are both decreasing $\sum a_n$ and $\sum b_n$ both ...
4
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2answers
57 views

Finding limit via Sandwich Theorem: $\lim_{n\to\infty} n\sum_{n+1}^{2n} \frac{1}{i^2}$

Question: Use the Sandwich Theorem to find $$\lim_{n\to ∞} n\sum_{n+1}^{2n} \frac{1}{i^2}$$ Appreciate any guidance.
2
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1answer
23 views

Can you prove this identity involving the divisor sum function?

Let $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisor function. I've deduced, but I don't know how prove directly $$\sigma(n)=\sum_{m=1}^{n}\sum_{k=1}^{m}(-1)^{nk}\cos\left(\frac{\pi ...
2
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2answers
37 views

Limit of the sequence $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$.

I have tried to solve this limit : $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$. Where n $\in\mathbb{N}$. I have understood that the limit exists and goes to 0 if the argument becomes ...
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0answers
12 views

$\pi_n(y)\in O(1)$ for every realisation $y$ of $Y_n$ implies $\pi_n(Y_n)\in O_p(1)$

Consider a sequence of random variables $\{Y_n\}_n$ all defined on the same probability space $(\Omega, \mathcal{F}, P)$ such that $Y_n:\Omega \rightarrow \mathbb{R}$ $\forall n$. Consider a sequence ...
1
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0answers
26 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
1
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1answer
31 views

Finding limit of $\frac{a_n^3+5n}{a_n^2+n}$ for $(a_n)$ bounded.

Suppose that the sequence $(a_n)_{n \in \mathbb{N}}$ is bounded. Prove that the sequence $(c_n)_{n \in \mathbb{N}}$ defined by $$ c_n = \frac{a_n^3+5n}{a_n^2+n} $$ is convergent and find its ...
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1answer
26 views

To check whether series S and T are convergent or not [on hold]

To check whether series S and T are convergent or not . I applied ratio test for series S and found it to be convergent but i do not know about series T. Thanks
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4answers
51 views

Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing?

Is the sequence $\sqrt{1+\frac{1}{n^2}}$ increasing or decreasing? I simplified it to $\frac{\sqrt{n^2+1}}{n}$, and I tried $a_{n+1}-a_n$ and $\frac{a_{n+1}}{a_n}$, but neither seem to work, how ...
0
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0answers
42 views

Find the Summation of Summation [on hold]

An Array A consisting of N integers .We perform the following operation M times: for i = 2 to N: Ai = Ai + Ai-1 We have to find xth element of the array ...
0
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0answers
19 views

Epsilon delta proofs of theorems of continuity

Can anyone suggest a book which contains epsilon delta prooves for properties and theorems of continuity rather than sequential proofs.
2
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1answer
28 views

Does this conjecture in 1-d real analysis seem reasonable

Hello I am currently trying to prove a result and I have basically whittled it down to showing the following is true. Let $I\subset\mathbb{R}$ be an interval and fix $\alpha>1$ real number. Fix ...
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2answers
47 views

Is the sequence $s_n= \frac{\sin\frac\pi2}{1\cdot 2}+\frac{\sin\frac\pi{2^2}}{2\cdot 3} + \dots + \frac{\sin\frac\pi{2^n}}{n\cdot(n+1)}$ convergent? [on hold]

Which of following is correct? I think option D. Not sure though $$s_n= \frac{\sin\frac\pi2}{1\cdot 2}+\frac{\sin\frac\pi{2^2}}{2\cdot 3} + \dots + \frac{\sin\frac\pi{2^n}}{n\cdot(n+1)}$$ is ...
1
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1answer
26 views

Harmonic progression sum

http://www.mathalino.com/reviewer/algebra/arithmetic-geometric-and-harmonic-progressions Please go to this link and see how they tell you to find the sum of harmonic prgression. However I am sure it ...
2
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1answer
22 views

“Convergence”/“Divergence” of $\prod_{n=1}^\infty (1 - \gamma_n)$

While trying to understand a proof of a result in an article, I stumbled upon the product $$\prod_{n=1}^\infty (1 - \gamma_n)$$ with $\gamma_n$ a real scalar belonging to $(0,1)$. I'm not really a ...
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3answers
78 views

Find the limit $\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)$

$$\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)\;=\;?\quad(n\in I) \\ \text{where $\lfloor\cdot\rfloor$ is the greatest integer function.}$$ This is what I ...
0
votes
2answers
23 views

At wich $n$ reaches the sequence its target value?

There are three parameters: $y_s=y[0]$ start value $y_t=y[n]$ target value $\alpha, 0>\alpha\leq1$ smoothness Starting at $y[0]=y_s$ the sequence is developed with this recursive formula: ...
5
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2answers
80 views

Evaluate $\sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}$

I am trying to re-learn some basic math and I realize I have forgotten most of it. Evaluate $$\sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}$$ Call the terms $S_n$ and the total sum $S$. $$S_n < ...
2
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1answer
33 views

Sum of $n6^{n}z^{n-1}$

I'm going mad calculating the sum $\sum_{n=0}^{\infty}n6^{n}z^{n-1}$. I proceeded in this way: $\frac{1}{z}\sum_{n=0}^{\infty}n(6z)^n$ and I'd like to figure out a geometric serie, but how can I take ...
0
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0answers
27 views

counting steps in Collatz Sequence [on hold]

I'm tring to code a Java code to find the collatz sequence of a given integer. In the given problem I have they've given $6 \rightarrow 6\: 3\: 10\: 5\: 16\: 8\: 4\: 2$ It has taken $14$ steps to ...
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2answers
47 views

How many sequences of rational numbers in $[0,1]$ exist?

I was talking with a friend of mine and we wonder how many sequences of rational numbers on $[0,1]$ there exists. My first attempt was to consider that every sequence like that must be a subset of ...
2
votes
1answer
41 views

Conjecture: $\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$

I conjecture: $$\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$$ Is this provable? If this is false, then can I have a function $f$ such that: $$\lim_{x\to\infty} ...
2
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3answers
71 views

Is it solvable to ask to find a recurrence relation for the following sequence 4,7,2,20,31,73,155,332,715,… ?

Is it solvable to ask to find a recurrence relation for the following sequence 4,7,2,20,31,73,155,332,715,... without any other information? If not, what would be the very least amount of information ...
0
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1answer
571 views

Does this numerical series have any special name?

I don't have enough background to find the answer for this on my own, so I am posting it here with the hope to get some pointers. Assume a descending sequence of K numbers $\{n_1, ..., n_K\}$ where ...
0
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1answer
32 views

How many points among them should we take to ensure that some two of them are less than the distance $1/5$ apart?

We are given a fixed point on a circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0,1,2,\ldots$ from it we obtain points with ...
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votes
1answer
70 views

Solve this puzzle? [on hold]

Given a number, the answer is a power of $2$. Given $1.000$ the answer is $16384$. Given $5.000$ the answer is $131072$. Can someone find a function, so given any number we can get the answer?
1
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1answer
22 views

Show that $S^1$ is complete

Define: $S^1 = \{(x,y): x^2+y^2 = 1\}$ Then wish to show $S^1$ is complete Attempt: Let $(z_n)$ be a Cauchy sequence on $S^1$, $(z_n) = (x_n, y_n)$ Then $S^1$ is complete if $\forall ...
1
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0answers
33 views

On a “coincidence” of two sequences involving $a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$

This was inspired by this post. Define, $$a_n = {_2F_1}\left(\tfrac{1}{2},-n;\tfrac{3}{2};\tfrac{1}{2}\right)$$ $$b_n = \sum_{k=0}^n \binom{-\tfrac{1}{2}}{k}\big(-\tfrac{1}{2}\big)^k$$ where ...
0
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0answers
16 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
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votes
2answers
10 views

Find Nth element where difference between elements is in AP [on hold]

Let I have a sequence as follows, $$a_1,a_2,a_3,a_4,a_5...$$ where $(a_2-a_1),(a_3-a_2),(a_4-a_3),... $ are in arithmetic progression. How can I find the Nth element ($a_N$) of the series ...
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0answers
15 views

Solving summation with incomplete gamma function

I solved an indefinite integral that gave me the following result: \begin{equation} f(u) = -\sum_{k=0}^{n}\binom{n}{k}\frac{\mathrm{sgn}(u)^{k+1}\, b^{n-k}}{2\, a^{\frac{k+1}{2}}}\, ...
0
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0answers
22 views

$\lim\limits_{K\to\infty} [n,K]$.

I've seen the following notation sometimes $$\lim\limits_{K\to\infty} [n,K]$$ And some people claim this is equal to $[n,\infty)$. They say, well $K$ keeps getting larger so for whatever $x\ge n$, we ...
0
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1answer
34 views

Why does $x_n^2$ converge to $x^2$ if $x_n$ converges to $x$?

Let $x_n$ be a convergent sequence converging to $x$ Then claim $x_n^2$ converges to $x^2$ I wish to use the definition to show this is the case. Recall $x_n \to x$ iff $\forall \epsilon > 0, ...
2
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2answers
56 views

Show $\forall \delta > 0, \exists n \in \mathbb{N}$ such that $\frac{1}{n} < \delta$

The question is in the title, but I have no idea how to solve it, so a few hints would be appreciated, thanks.
0
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1answer
28 views

Convergence of a series depending on a parameter

I have the following series $$\sum_{n=2}^{\infty} \frac{n}{(n-1)^2+\alpha 2^n}$$ I have to find for which $\alpha$ this series converges. I tried the ratio test but I get $\lim_{n \to \infty} ...
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5answers
38 views

Prove $a_t \rightarrow x$ using the Betweenness Property

Prove that for any $x \in \Bbb R$ there is a strictly increasing sequence ($a_t$) in $\Bbb Q$ such that ($a_t$) converges to $x$, (i.e. $a_t \rightarrow x$) I want to prove this using the Betweenness ...
5
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1answer
35 views

$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$ - show that $a_n$ is convergent sequence

Problem: Show that $a_n$ is convergent sequence and find a limit of $a_n$. $$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$$ I tried to look at this as normal limit problem so I wrote ...
8
votes
2answers
83 views

$p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$; $\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}} = ? $

Let $p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$ The sum $$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}}$$ Can be expressed as $\frac{a}{b}$. Where $\lfloor \cdot \rceil$ denotes the ...
4
votes
3answers
340 views

Is there any partial sums of harmonic series that is integer?

is there any partial sums of harmonic series that add up to an integer? partial sums not as trivial as the first term only i.e. 1, or the powers of 2 i.e the infinite geometric series for 2. This ...
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votes
0answers
17 views

Solution to recursive equation

what will be the form of solution for this kind of recurrence equation? $$P_{n+1} + \dfrac{2n P_n}{x} - P_{n-1} = 0$$ $x$ is a constant. Will a guess solution of form $\lambda^n$ work? I need to ...
0
votes
2answers
59 views

Finding an expression for the sum of n tems of the series $1^2 + 2^2 + 3^2 + … + n^2$ [duplicate]

Possible Duplicate: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$ I know that if you have a non-arithmetic or geometric progression, you can find a sum $S$ of a ...
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votes
2answers
169 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
1
vote
1answer
48 views

If $\sum\limits_n \sqrt{\mathrm{Var}(X_n)}$ converges, then $\sum\limits_n (X_n - E[X_n])$ converges in $L^2$

I have seen in a paper to claim the following: If $\sum\limits_n \sqrt{\mathrm{Var}(X_n)} < \infty$, then $\sum\limits_n (X_n - E[X_n])$ converges in $L^2$, for any sequence of random ...
2
votes
2answers
50 views

Comparing a Factorial and a Perfect Power

Let us define the following recurrence relations as so. $$a_1=6, a_{n+1}=a_n!$$ $$b_1=6, b_{n+1}=6^{b_n}$$ So, which of the following is larger? $a_{b_2}$ or $b_{a_2}$? To clarify, I am trying to ...