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

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3
votes
2answers
31 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.
0
votes
0answers
12 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
votes
2answers
35 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 ...
-1
votes
0answers
11 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
vote
0answers
23 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
vote
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 ...
-2
votes
1answer
24 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
1
vote
4answers
49 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
votes
0answers
36 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
votes
0answers
17 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
votes
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 ...
-3
votes
2answers
46 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
vote
1answer
25 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
votes
1answer
21 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 ...
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
votes
2answers
78 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
votes
1answer
24 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
votes
0answers
25 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 ...
-1
votes
2answers
41 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
39 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
votes
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
votes
1answer
538 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
votes
1answer
28 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 ...
-2
votes
2answers
52 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
vote
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
vote
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
votes
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 ...
-1
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 ...
0
votes
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
votes
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
votes
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
votes
2answers
55 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
votes
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} ...
1
vote
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
votes
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
79 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
338 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 ...
-1
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
58 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 ...
-6
votes
2answers
157 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
46 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 ...
0
votes
1answer
16 views

Why the dual to $c_0$ is $l^1$ and the space of sequences with bounded partial sums?

The dual to $c_0$ is $l^1$, but if $\{x_n\}_{n\in\mathbb{N}}\in c_0$, than according to Dirichlet's test $\sum_{n\in\mathbb{N}}(-1)^nx_n$ converges. But $\{(-1)^n\}_{n\in\mathbb{N}}\notin l^1$. So why ...
0
votes
0answers
15 views

How could I prove that equivalence on limits of sequences?

$T$ is a positive real number $\lambda_k$ is a family of positive reals number that $\to \infty$ with $k$ $f(s)$ is a function that is a $o(s)$ and $0<f(s)<s$ (these properties may be ...
0
votes
2answers
26 views

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n\}$ is closed

Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n \}$ is closed. I had to show it is compact, and I am done showing it is relatively compact, but now I am stuck showing it is closed. ...
6
votes
0answers
62 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
0
votes
1answer
56 views

Limit of a sequence with binomial coefficient. Can I use Stirling?

I was trying to solve this limit: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $ I solved it with Cesaro theorem: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $= $\lim_\limits{n\to \infty} ...
0
votes
2answers
23 views

Calculating limit of a series of series

Given is the following series I want to calculate the limit for $n \to \infty$. I already recognised the geometric series $\sum_{i=2}^n a^{i} = \frac{1}{1-a}$ for $a=e^\rho$ (since rho is ...
0
votes
0answers
23 views

What kind of information does the derivative of a sequence of functions give us?

When we derivate a function we know for example when then function increases and drecreases but when I derivate a sequence of functions I don't how to interpret the derivative since it is a collection ...
1
vote
2answers
24 views

Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...