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

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1answer
37 views

$\{na_{n}\} \subset \ell^{\infty}(\mathbb Z)$ if $a_{n}\in \ell^{1}$?

Suppose $\{a_{n}\} \subset \ell^{1}(\mathbb Z)$ (Sequence space). MY Question: Can we expect $\{na_{n}\} \subset \ell^{\infty}(\mathbb Z)$?
3
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1answer
25 views

Does $(\ell^{1}(\mathbb Z), \cdot)$ have a bounded approximate identity?

Put $\ell^{1}(\mathbb Z)=\{f:\mathbb Z \to \mathbb C: \|f\|_{\ell^{1}}:=\sum_{n\in \mathbb Z}|f(n)|< \infty \}$ and we note that $\ell^{1}(\mathbb Z)$ is an algebra under pointwise multiplication. ...
0
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1answer
39 views

Finding a positive lower bound of hte sequence $\frac{\sqrt[n]{n!}}n$

I am given a sequence {(n'th root of n!)/n}. Can I show that the sequence is bounded below by a real no. which is greater than 0, by not calculating the limit of it....???thank you
2
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0answers
53 views

Prove that this infinite product converges uniformly,

Let $(a_n)_{n=1}^\infty$ be a sequence of complex numbers such that (i) $0<|a_n|<1$, (ii) $\sum_{n=1}^\infty(1-|a_n|)<\infty$ Prove that the infinite product $$\prod_1^\infty \frac{(a_n ...
3
votes
1answer
74 views

Finding a power series representation for $\left(\frac{x}{2-x}\right)^3$

Find a power series representation for $\displaystyle\left(\frac{x}{2-x}\right)^3$ My approach is in finding something similar to $\displaystyle\left(\frac{x}{2-x}\right)^3$ to which I can easily ...
0
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1answer
44 views

Is there a theorem relating sequences to its series or vice versa?

I only have these in mind Theorem: If a series $\sum_n a_n$ of real numbers converges then $\lim_\limits{n \to \infty} |a_n|=0$ Divergence test: If $\lim_\limits{n \to \infty} a_n \neq 0$, then the ...
2
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1answer
32 views

Correctness of proof for the convergence of a series

Does the following series converge? $\sum_{n=1}^\infty \frac{5^n + (-1)^n}{2^n3^n}$ What I've done so far. $0\leq \sum_{n=1}^\infty \frac{5^n + (-1)^n}{2^n3^n} \leq \sum_{n=1}^\infty \frac{5^n + ...
1
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1answer
32 views

can you consider a series to be a sequence of sums?

for example sequence: $1/(2^n), \qquad n\ge 0$ sequence for the series: $1, 1.5, 1.75, 1.875, \ldots$ and if so, does that mean you can use/extend sequence theorems for series?
0
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1answer
49 views

Why does the limit not change in the given summation series even after substituting $p=-n$ in the given question?

$$ \begin{align} DFS[x^*(-n)] &= \frac{1}{N}\sum^{N-1}_{n=0}x^*(-n)e^{-j2\pi kn/N}\\ &= \left[\frac{1}{N}\sum^{N-1}_{n=0}x(-n)e^{j2\pi kn/N}\right]^*\\ &= ...
13
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3answers
228 views

What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial ...
1
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1answer
29 views

Find Closed-form expression of a integer sequence.

We have a integer sequence $u_{n+1}=2p.u_{n}-u_{n-1}$ (p is a positive integer, $n\geq 3$ ) When $u_{1}=1; u_{2}=p$ then with a regular way I can find Closed-form expression of the sequence. ...
1
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3answers
185 views

How to calculate the limit of this sum with different methods? [duplicate]

It's a basic question , but what are the common methods to calculate limits like this one: $$\sum_{k=1}^\infty \frac{3k}{7^{k-1}}$$
2
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6answers
128 views

How to prove $\lim_{n\rightarrow\infty}nx^n=0$ without L'Hôpital's rule, where $x \in [0,1)$??

How to prove $$\lim_{n\rightarrow\infty}nx^n=0$$ without L'Hôpital's rule? where $x \in [0,1)$ and $n=1,2,3,...$. I know one of way to prove this is to treat $n$ is real, and $n$ and ...
3
votes
3answers
80 views

Prove that the sequence $a_n=\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}$ is monotonically decreasing sequence

Prove that the sequence $$a_n=\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}$$ is monotonically decreasing sequence. I tried $a_{n+1} - a_{n} < 0$, but i was not able to do it. Help ...
1
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0answers
16 views

Representing ordered sequence as a vector

What is the best way to represent a bunch of ordered sequence as a vector in a d-dimensional space? Imagine we have some ordered sequences like this: ...
0
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0answers
30 views

Sum of exponential formula [duplicate]

If, $\sum_{n=-\infty}^\infty e^{-n^2\pi x}=f(x)$ How can it be proven that $f(x)=\frac{1}{\sqrt x}f(\frac{1}{x})$?
2
votes
2answers
64 views

Given $x_0>0$ and $a>0$, if $x_{n+1}=\frac{2a^2x_n}{x_n^2 + a^2}$ for every $n\ge0$ then $\lim\limits_{n\to\infty}x_n=a$

Given $x_n$ be a sequence of positive numbers such that $$x_{n+1}=\frac{2a^2x_n}{x_n^2 + a^2} , a>0.$$ Show that $$\lim_{n\to\infty}x_n=a.$$ I thought of first proving sequence bounded and ...
1
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2answers
61 views

Rearrangment of convergent series

Consider the convergent series $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots \tag{$*$}$$ and one of its rearrangments ...
2
votes
3answers
54 views

Deducing $\sum_{r=1}^{n}r$ from sine summation formula

We know the famous formula $$\sum_{r=1}^{n}\sin r\theta=\sin \frac{n\theta}{2}\csc\frac{\theta}{2}\sin\frac{(n+1)\theta}{2}\ .$$ I have come across a question that use the above result to find ...
2
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3answers
63 views

Convergence and Limit of a Sequence

I am looking at the sequence and I'm trying to see what happens as $n\to\infty$ $$ a_n = \frac{\prod_{1}^{n}(2n-1)}{(2n)^n} = \frac{1\cdot3\cdot5\cdot...\cdot(2n-1)}{(2n)^n} $$ By inspection, I can ...
3
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1answer
78 views

Can we say $(\prod_{n=1}^{\infty}n)^2=(\prod_{n=1}^{\infty}n^2)$

Can we say $$(\prod_{n=1}^{\infty}n)^2=(\prod_{n=1}^{\infty}n^2)$$ It works fine if things are finite, does it hold in $n$ goes to infinity?
0
votes
4answers
272 views

How to show this limit is zero?

Is it true that the following limit is zero as $n$ goes to infinity for all positive integers $k$? If so, how to prove it? $$n\left[(n-1)^{-\frac{1}{k}}-n^{-\frac{1}{k}}\right]$$
7
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0answers
82 views

Infinite Integration in Limits of Integration

Given the following: $$ u_0 = \int \limits_{ 0 } ^{ 1 } x \, dx , \:\:\: u_1 = \int \limits^{ \int \limits_{ 1/2 } ^{ 1 } x \, dx } _{ \int \limits_{ 0 } ^{ 1/2 } x \, dx } x \,dx , \:\:\: u_2 = \int ...
1
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0answers
39 views

Convergence and limit of a sequence $x_n=1+\frac{x^2_{n-1}}{2},n\ge2,x_1=\frac{3}{8}$

$x_{n+1}-x_n=\frac{x^2_n-2x_n+2}{2}>0$ sequence is increasing. I don't know how to prove that it is bounded. Limit should be $\frac{1}{2}$
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2answers
30 views

Find the value of x if 1,$\log_{9}(3^{1-x}+2) $, $ \log_{3}(4.3^x-1) $ are in an AP.

We have an AP: 1, $ \log_{9}(3^{1-x}+2) $, $ \log_{3}(4.3^x-1) $ We have to find value of x. $$ d = a_{2} - a_{1} $$ $$ d = log^{(3^{1-x}+2)}_9 - 1 $$ $$ d = log^{(3^{1-x}+2)^{\frac{1}{2}}}_3 - ...
14
votes
6answers
3k views

Is this solution mathematically “legal”?

I have the sequence $$ a_n = \frac{n \cos n}{n^2 + 1} $$ and I'm trying to evaluate the limit of $a_n$ as $n\to\infty$ $$ \begin{align*} \lim_{n\to\infty}a_n&= \lim_{n\to\infty}\frac{n \cos n}{n^2 ...
1
vote
1answer
50 views

Sequence identification for these numbers?

Sorry for my English. Can someone help me find the generating formula for these numbers? $$[255,2915,16383,62499,186623,470595,1048575,2125763,3999999,7086243,11943935,19307235,...]$$ All I know is ...
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0answers
54 views

Question re Hold'em Hands with Identical Ranks [duplicate]

In a prior posting regarding a poker session consisting of 142 hands I did not clearly state my question resulting in a misinterpretation. Let me try again. Out of the 142 hands dealt, there were 38 ...
0
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1answer
29 views

A question on convergent series of positive terms

Let $\sum a_n$ be a convergent series of positive terms ; then we know $\lim \inf (na_n)=0$ ; can we derive from here that if $\{a_n\}$ is decreasing , then $\lim (na_n)=0$ ?
2
votes
4answers
141 views

Find $\lim\limits_{n\to+\infty}\sqrt[n]{\prod\limits_{k=1}^{n}{n\choose k}}$

I tried to compute the product of binomial coefficients. I found that $$\prod\limits_{k=0}^{n}{n\choose k}=\frac{H^2(n)}{(n!)^{n+1}}$$ I am not familiar with hyperfactorial function. How to find ...
1
vote
3answers
116 views

Sequence defined by $s_{n+1}=\sqrt{s_{n}s_{n+2}}$

Let $(s_n)$ be the sequence defined by: $$ s_0,s_1\in \mathbb{R}^{+},\quad \forall n\in \mathbb{N};\quad s_{n+1}=\sqrt{s_{n}s_{n+2}} $$ $(s_n)$ is arithmetic sequence $(s_n)$ is ...
2
votes
0answers
49 views

Optimizing over an infinite set of variables

This may be a very basic question, but it's been a while since I did any optimization. Suppose I have a sequence $(x_i)$, $i=1,2,\ldots$ in the $\ell^2$ space and the following optimization problem: ...
1
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2answers
70 views

Series Approximation How to evaluate $1/3+1/3(1/3)^3+1/5(1/3)^5+…$?

How to evaluate $$\frac13+\frac13(\frac13)^3+\frac15(\frac13)^5+...$$? I faced this particular sum in the website www.toppr.com .And it is given under the heading "Problems on Approximation"...but I ...
0
votes
1answer
53 views

Find an expression for the sum of the first n terms of this series

I want to find an expression for the sum of the first n terms of this series: \begin{equation} 5-\frac{5}{2}+\frac{5}{4}-\frac{5}{8}+...+\frac{(-1)^{n-1}5}{2^{n-1}} \end{equation} I have proved that ...
2
votes
3answers
153 views

How to solve this double summation?

$$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{3^{mn}}$$ And, $$\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}\frac{1}{3^{mn}}$$
1
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2answers
49 views

Solve the sequences inequality

If $a_1=1$ and $a_n=a_{n-1}+\dfrac{1}{a_{n-1}}$ for $n≥2$ , then prove that $12 < a_{75} < 15$ ? I have tried solving this by: $$a_{75} - a_1 = ...
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2answers
25 views

Convergent series? The pairwise product of the harmonic series with a nonnegative, diminishing to zero, divergent series

Suppose we have a sequence $\{\alpha_n\}$ such that $\alpha_n \ge 0$, $\sum_n \alpha_n = \infty$ and $\alpha_n \rightarrow 0$. Is it true that $\sum_n n^{-1} \, \alpha_n$ converges?
7
votes
2answers
120 views

Bounding $f'$ in terms of $f$ and $f''$

Assume that $f: \mathbb{R} \to [0,\infty)$ is $C^2$ and $|f''(x)| \leq A$ for all $x$. Show that the inequality $$(f'(x))^2 \le 2Af(x)$$ holds for all $x$. The hint given in the question was, ...
0
votes
1answer
65 views

Partially Identical Hands in Hold'em

I was playing Texas Hold'em at a local cardroom last night keeping a meticulous record of the hands I was dealt. Perhaps I am totally wrong but I thought the occurrences of certain events in this ...
7
votes
2answers
235 views

What is the probability that this harmonic series with randomly chosen signs will converge?

Suppose we fix $p$ between $0$ and $1$ (without loss of generalization, we can assume $p \leq 1/2$). Then suppose we form the series $\sum_n a_n / n$ where the $a_n$ are independent random variables ...
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2answers
119 views

a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
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5answers
132 views

Convergent/divergent series

Is the following series divergent/convergent? ...
0
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0answers
52 views

How to prove that this series converges. [duplicate]

I want to prove that this series converges: \begin{equation} 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+... \end{equation} I normally use this one as a standard series to test other series for their ...
8
votes
3answers
73 views

Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?

Let $P$ be the set of primes $p$ greater than $3$ such that $p\equiv1 \pmod{4}$. Does the following sum converge or diverge? $$ \sum_{p\in P}\frac{1}{p} $$
2
votes
4answers
50 views

For what values of $p$ does this series converge?

This is a question we asked on a second semester calculus test. For what values of $p$ does this series converge? $$\sum_{n=1}^{\infty}\frac{\sin(1/n)}{n^p}$$ I believe that it actually can be shown ...
0
votes
1answer
31 views

Example of each $\{f_n\}$ Riemann integrable such that $\sum f_n$ converges point-wise to $f$ which is not Riemann-integrable

I am looking for an example of a sequence $\{f_n\}$ of real valued Riemann integrable functions on a closed bounded interval such that $\sum f_n$ converges point-wise to a function $f$ which is not ...
0
votes
0answers
27 views

Limit Points and Convergence of Sequences.

Let $E \subseteq \mathbb{R}$ (or $\mathbb{C}$). A point $p \in \mathbb{R}$ (or $\mathbb{R}$ ) is called a limit point of $E$, if $\forall \epsilon > 0$, $\exists z \in E$ such that $0 < |z − p| ...
1
vote
1answer
33 views

Transforming a sequence to distinguish a limit

This might be the wrong place to ask this question, but I figured I might get some creative answers: I have a decreasing sequence $\{a_n\}_{n \geq 1}$ with $a_k \in (0,1)$ for all $k$ and $a_n \to ...
3
votes
2answers
214 views

Can you use the sum formula for a geometric series starting at any point?

Wherever I see the sum of a infinite geometric series with $|r|<1$ being derived the series always starts at $n = 0$, or $n = 1$, the basic form is $$a + ar + ar^2 + ar^3 + ... $$ And the sum is ...
0
votes
2answers
49 views

If $\,S_n/n\,$ converges to a finite limit $c$, why does $\,\left(S_{n+1}-S_{n}\right)/n\,$ converge to zero?

If we have a sequence $S_n$ and know as a fact that $S_n/n$ converges to some finite limit $c$, why is it true that $(S_{n+1}-S_n)/n$ converges to zero? I could see this for $n+1$ in the denominator ...