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

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14 views

Find the Summation of Summation

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 ...
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0answers
11 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
19 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
32 views

Which of following is correct for this sequence

Which of following is correct ? i think option D . Not sure though
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1answer
24 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
18 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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2answers
22 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
77 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
22 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 ...
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0answers
23 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
37 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
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1answer
35 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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2answers
49 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 ...
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1answer
403 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 ...
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1answer
22 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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2answers
49 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?
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1answer
21 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 ...
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0answers
30 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 ...
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0answers
15 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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2answers
9 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}}}\, ...
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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 ...
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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, ...
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2answers
53 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.
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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
37 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
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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
335 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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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
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2answers
56 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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2answers
138 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 ...
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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
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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 ...
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1answer
15 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 ...
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0answers
14 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 ...
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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
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0answers
57 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 ...
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1answer
47 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} ...
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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 ...
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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 ...
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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 ...
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1answer
78 views

Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

Having $f(z)=\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n$ I had to find what $\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n\sum\limits_{n=0}^{+\infty} \frac{D_n}{n!}z^n=\sum\limits_{n=0}^{+\infty} ...
0
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2answers
23 views

How do I use the ratio test to determine convergence or divergence in this problem?

I have the problem: $$a_{n} = \frac{e^{n+5}}{\sqrt{n+7}(n+3)!}$$ I am told to use the ratio test to determine convergence or divergence (or the test could be inconclusive). So I take the limit: ...
1
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3answers
61 views

Solve the following sequence problem

Let a sequence be defined as $$a_n=\lim_ {x \to 0}{1-\cos (x)\cos(2x).....\cos (nx)\over x^2}$$ a)prove that the given sequence is monotonic and that it is not bounded above. b)calculate $$\lim_{n \to ...
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0answers
8 views

Lagarias and Robin theorems versus multiplicative property

If I use for example Robin's theorem, see here in the section Growth of arithmetic functions, or Lagarias equivalence, see (5) here has sense ask us what is the more sharp inequality for ...
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2answers
45 views

$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$ iff $\lim_{n\to\infty}{(a_{n}-b_{n})=0}$ [on hold]

I need to proof or disproof that: $$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$$if and only if $$\lim_{n\to\infty}{(a_{n}-b_{n})=0}$$
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0answers
16 views

Express $\prod \frac{a_i}{x+b_i}$ in terms of known functions [on hold]

I am interested if there is a way to express the finite product $$ \prod_{i=1}^{S} \frac{a_i}{x+b_i} $$ in terms of known functions like Gamma, Beta, etc.
0
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1answer
24 views

Why is the following reverse triangle inequality true for given series?

I wish to show that for $(a_k)$ a sequence of numbers, $a_k \in \mathbb{R}$ then claim : $|\sum\limits_{k = n+1}^m a_k | \leq ||\sum\limits_{k = n+1}^\infty a_k| - |\sum\limits_{k = ...
6
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0answers
47 views

Variation of the Kempner series

It is easy to argue that the Kempner series converges: $$ \sum\limits_{\substack{n \text{ : 9 is}\\\text{ not a digit of } n}} \frac{1}{n} < \infty$$ Let $E \subset \Bbb N_{>0}$ the subset ...