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

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20
votes
4answers
929 views

Elementary Proof of Euler's Sine Expansion $\prod_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$

I've been looking at proofs of Euler's sine expansion, that is $$\frac{\sin x}{x}=\prod_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$$ All the proofs seem to rely on complex analysis and ...
108
votes
9answers
11k views

Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?

If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer. If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ ...
-1
votes
0answers
20 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
1answer
48 views

Convert Riemann sum to definite integral: $\sum_{i = 1}^n \frac{n}{n^2 + i^2}$

I am having trouble with this problem. Basically, I am given a Riemann sum and I have to rearrange it so that I can deduce the definite integral that it is equivalent to. Thank you. $$\lim_{n \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 ...
1
vote
1answer
51 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
55 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
17 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
2answers
29 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. ...
0
votes
0answers
17 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
29 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 ...
6
votes
0answers
79 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 ...
1
vote
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} ...
2
votes
1answer
46 views

If a sequence of functionals converges weakly then it is bounded.

Let $f_k, f \in L^{\infty}(R)$ and $f_k \overset * \to f$ in $L^{\infty}(R)$. Is $f_k$ a bounded sequence in $L^{\infty}(R^n)$? (Definition: if $(v_n)$ is a sequence in $V = X^*$, we say that $v_n ...
-1
votes
2answers
46 views

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

I need to proof or disproof that: $$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$$if and only if $$\lim_{n\to\infty}{(a_{n}-b_{n})=0}$$
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 ...
23
votes
4answers
836 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
1
vote
3answers
63 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 ...
1
vote
2answers
25 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 ...
0
votes
2answers
25 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: ...
0
votes
0answers
9 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 ...
11
votes
4answers
504 views

Finding the power series for $y$ where $y + \sin(y) = x$

What do you do to find the power series for an inverse relationship such as for $y$ in $y + \sin(y) = x$? I'm not sure where to begin. (Similarly, the Lambert $W$ function has such a power series ...
0
votes
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 = ...
-1
votes
0answers
17 views

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

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.
5
votes
1answer
115 views

How do you find the probability of A winning if the probability of getting a favourable outcome in the $r^{th}$ turn is a function of $r$?

Problem: Two players A and B are playing snake and ladder. A is at 99 and he needs 1 to win in rolling of a dice. However, he is always allowed to re-throw the dice if 6 appears. What is the ...
3
votes
2answers
20 views

Arithmetic Sequences - accountant problem.

I need help with $part$ $c$ of the following problem: An accountant has a salary scheme as outlined below. STARTING SALARY £16,000. ANNUAL INCREASES OF £1,000 GUARANTEED THEREAFTER. Let £$u_n$ ...
0
votes
0answers
41 views

How can I change a summation of cosines to a product of cosines for higher degree functions?

I was wondering if I can get an alternate form of a sum over cosines $$\sum_{n=1}^m\cos{f(n)}$$ and I found that I can. We must however make a modification to the upper limit with $m\rightarrow2^m$. ...
0
votes
1answer
41 views

Infinite Sum Defined by $\int \frac{e^x}{x}dx$ vs. Exponential Function Taylor Series

Recently, when fiddling around with integration by parts, I noticed that it is possible to define infinite series that led to an integral. My calculus teacher noticed this, and told me to find $$ ...
2
votes
1answer
74 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
2
votes
3answers
241 views

Showing that this sequence is eventually decreasing

I'm trying to show that this sequence $$a_n = \frac{3^n-7}{4^n+5}$$ is decreasing for all $n$ greater than some $N\in \Bbb N$. All I can see to do is something like $$a_{n+1} = ...
1
vote
1answer
18 views

$a_n\geq b_n$ for $n>\bar{n}$ implies $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$

Consider two sequences of real numbers $\{a_n\}_n, \{b_n\}_n$. I know that if $a_n\geq b_n$ $\forall n$ then $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$. Suppose ...
0
votes
1answer
39 views

Find the range of $x$ for which the sequence $\dfrac{n!} {k!(n-k)!}x^n $ converges to $0$ for a stabilised $k\in\mathbb{N}$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $c$ and graded for ...
4
votes
3answers
103 views

Limit of $a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$

Find a limit of sequence: $$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$ $$a_1=0,a_2=0$$ I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by ...
0
votes
0answers
26 views

Can you define a Cauchy product from this identity?

I can write vague expessions for $\zeta(3)$, the Apéry's constant, for example when I multipliy by $\frac{1}{n^4}$ the recursion relation (2) in page 2 here, and after I take the sum ...
1
vote
2answers
129 views

Understanding Sobol sequences

Can someone explain to me in simple terms, how Sobol sequences work? The wikipedia article is fairly technical. They look pretty interesting. So I shall describe (whatever little I know) in short the ...
0
votes
1answer
32 views

Is there any identity for this series?

While solving inequality and finite series problem I often come across this series- $$(n+1)(n+2)(n+3)...(n+n)$$. Is there a general solution to this form of a series? Thanks for any help!!
0
votes
3answers
38 views

Arc length of a sequence of semicircles.

Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$. Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one ...
2
votes
2answers
42 views

Can you get a closed-form for $\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!}$?

Let $B_{k}$ the kth Bernoulli number, then using their asymptotic I can justify the absolute convergence of this series $$\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!},$$ since, if there are no ...
1
vote
3answers
36 views
1
vote
1answer
40 views

Error with the proof that all solutions to the Cauchy Functional Equation are linear

If $f(x)$ is continuous, it is known that $f(x+y)=f(x)+f(y)$ implies that $f(x)$ is linear, and non-continuous solutions are discussed in these links. (1, 2,3, 4) However, what is wrong with this ...
7
votes
3answers
164 views

Variety of proofs of a simple proposition in arithmetic

I have a couple of simple proofs of a simple proposition and I'm curious to see the variety of different approaches others would take to prove the same thing. Definition: The dilation of a sequence ...
3
votes
2answers
75 views

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero?

Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero? I was talking to a friend about it but neither of us were able to come up with anything. We're not sure if $\pi$ is essential ...
0
votes
1answer
29 views

Bounded away sequence implications

Consider the sequence $\{\sqrt{n}|a_n-a|\}_n$ where $a_n, a \in \mathbb{R}$. Assume $\{\sqrt{n}|a_n-a|\}_n$ is bounded away from $0$ and $\infty$. Is this equivalent to or sufficient or necessary for ...
0
votes
2answers
23 views

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,…$

Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,...$ I've tried to isolate n but its impossible to do.
5
votes
7answers
151 views

Convergence of the recursive sequence $a_{n+1}=\sqrt{2-a_n}$

I'm attempting to prove the convergence (or divergence, though I strongly suspect it converges) of a sequence defined as $a_{n+1}=\sqrt{2-a_n}$ with $a_1=\sqrt{2}$. I cannot use the monotonic ...
1
vote
1answer
20 views

Implications of $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$

Consider two sequences of real numbers $\{a_n\}_n$, $\{b_n\}_n$. Suppose $n a_n=O((n^{\frac{1}{\alpha}}|b_n|)^\alpha)$ where $\alpha \in \mathbb{R}$, $na_n\geq 0$ and big $O$ notation is explained ...
-2
votes
1answer
35 views

How to get from left to right-hand side of the equation? $ \sum_{k=0}^{d} \binom{2d+1}{k} = \frac{1}{2} \cdot 2^{2d+1} $

I would like to know how the left hand side of the equation is achieved. In particular why the $\frac{1}{2}$ is there. I don't understand how one can get from the left to the right side. $$ ...
2
votes
2answers
110 views

Diagnosing essential Classical Mathematical Analysis I knowledge needed for II

I need to take Classical Mathematical Analysis II (Chapters 7-10: Sequences & Series of Functions, Special Functions (Exp/Log/Fourier/Gamma), Functions of Several Variables, Integration of ...
1
vote
2answers
30 views

Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,…

I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a ...
0
votes
1answer
13 views

Geometric progression with reverse order

I have the following problem: Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$. Thank you!