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

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1
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2answers
70 views

Limit of a Riemann sum $\lim_{n\to \infty} \frac{1^p+2^p+\ldots+n^p}{n^{p+1}}$

I have$$\lim_{n\to \infty} \frac{1^p+2^p+\ldots+n^p}{n^{p+1}}=$$ I managed to simplify it down to $$=\lim_{n\to \infty}\left( \left(\frac1n\right)^p \cdot \frac1n + \left(\frac2n\right)^p \cdot ...
0
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1answer
21 views

Condition for derivative sequence to converge?

Let $E_n(R)$ be a sequence of function that converge to $E(R)$, When we can said that $\dot{E}_n(R)$ converge to $\dot{E}(R)$. I can assume: uniform converge of $E_n$ to $E$. $E(R)$ convex. ...
7
votes
3answers
587 views

How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?

How to find the value of $S(\infty)$, where $S(n)$ is the following $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
1
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0answers
80 views
+50

Cauchy product $\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}a_{n-k}b_k$

I have been told that, if $\{a_n\}_{n\in\mathbb{N}}$, $\{a_{-n}\}_{n\in\mathbb{N}^+}$, $\{b_n\}_{n\in\mathbb{N}}$ and $\{b_{-n}\}_{n\in\mathbb{N}^+}$ are absolutely summable complex sequences, ...
1
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4answers
22 views

Function as a series :

Let $f(x)=\sum_{n=0}^{+\infty}\dfrac{x^n}{n!}$. Verify that $$\int_0^xf(x)dt=f(x)-1$$ This is the exercise 3 of the section $7.4$, of Guidorizzi's Calculus, Vol. 4. What I have tried: By the ratio ...
5
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3answers
98 views

Limit of the sequence $\frac{1^k+2^k+…+n^k}{n^{k+1}}$ [duplicate]

How would someone find the limit of the sequence $a_n = \frac{1^k+2^k+...+n^k}{n^{k+1}}, k \in \mathbb{N}$ as $n$ goes to Infinity? Can someone give me maybe a hint where to start?
3
votes
1answer
29 views

Proving Archimedes Sequences Equal $\pi$.

I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
0
votes
1answer
23 views

How to know if a space has a convergent subsequence?

So this is something I have been struggling with lately... how do we generally know that a space/set has a subsequence that converges? The current one I am struggling with is the space of sequences ...
0
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1answer
31 views

Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
7
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2answers
56 views

Closed form of $\sum_{n=1}^{\infty} \frac{J_0(2n)}{n^2}$

I'm new in the area of the series involving Bessel function of the first kind. What are the usual tools you would recommend me for computing such a series? Thanks. $$\sum_{n=1}^{\infty} ...
0
votes
1answer
21 views

Geometric series for values between 0 and 1

I am given that geometric series is defined as the following $1-x+x^2-x^3+x^4$ for values in range $0<x<1$. I am also told expected value can be calculated by using the following equation: ...
0
votes
3answers
52 views

Does this sequence diverge to ∞?

The sequence $(a_n)_{n \geq 1}$ is defined as follows: $$a_n:= \begin{cases} 0 \quad \text{if} \quad n \quad \text{is odd}\\ n \quad \text{if} \quad n \quad \text{is even}\end{cases} \quad .$$ Does ...
1
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2answers
18 views

Convergence of a sequence of functions involving rational and irrational numbers

For each $n\in \mathbb N$, let $f_n(x)=lim_{m\rightarrow \infty} (cosn!\pi x)^{2m}$ Show that the sequence converges on $\mathbb R$ to the function $f$ defined by $$f(x)=1, x\in \mathbb Q$$ $$=0, ...
5
votes
2answers
75 views

Computing in closed form $\sum_{n=1}^{\infty}\frac{\operatorname{Ci}\left(\frac{3}{4}\zeta(2) \space n\right)}{n^2}$

What tools would you recommend me for computing the series below? $$\sum_{n=1}^{\infty}\frac{\operatorname{\displaystyle Ci\left(\frac{3}{4}\zeta(2) \space n\right)}}{n^2}$$ I lack the starting ...
5
votes
3answers
32 views

Convergence of a series with general term $u_n=\int_0^{\infty}e^{-x^n}dx$

I would like to find if the series $\displaystyle \sum_{n=1}^{\infty}u_n$ is convergent or divergent where $$u_n=\int_0^{\infty}e^{-x^n}dx. $$ I've tried to obtain $v_n$ with explicit form such that ...
5
votes
2answers
196 views

Divergence of $\sum\limits_{n=1}^{\infty} \frac{\cos(\log(n))}{n}$

I'm struggling to prove that $$\sum \limits_{n=1}^{\infty} \frac{\cos(\log(n))}{n}$$ diverges. Does anyone have any idea how to prove it? Breaking it into smaller pieces din't work. Maybe I should ...
2
votes
9answers
237 views

How can one prove $\lim \frac{1}{(n!)^{\frac 1 n}} = 0$?

I have tried bounding the terms by $\dfrac 1 {2^{\frac 1 n}}$, but this clearly cannot be made as small as possible.
22
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4answers
2k views

Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$ So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
1
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5answers
105 views

Does $\sum_{n=1}^\infty \frac{\cos(\ln(n))}{n}$ converge? [duplicate]

EDIT: the question is answered here Divergence of $\sum\limits_{n=1}^{\infty} \frac{\cos(\log(n))}{n}$ Using integrals, I managed to prove that $$\displaystyle \forall m, \sin(\ln(m+1))\leq ...
2
votes
1answer
48 views

How to find this double summation?

To find the value of $$\sum_{m=1}^{∞}{\sum_{n=1}^{∞}{\frac{m^2\cdot n}{3^m \cdot (n\cdot 3^m+m\cdot3^n)} } }$$ I dont know how to proceed to these kind of problems. Can anybody provide a sol to this ...
1
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0answers
19 views

Prove that this sequence of fractions is returned unchanged after the divisor recurrence, the matrix inverse, and the sum over divisors.

Consider the fraction of binomial coefficients and powers of $4$: $$a(n)=\frac{\binom{2 (n-1)}{n-1}}{4^{n-1}}$$ starting: ...
11
votes
3answers
416 views
+300

Is there an elementary proof for Euler's product for Sine?

I've been looking at proofs of this equation: $$\displaystyle \frac {\sin \pi x} {\pi x} = \displaystyle \prod_{k \mathop = 1}^\infty \left({1 - \frac{x^2}{k^2} }\right)$$ All the proofs seem to ...
7
votes
7answers
343 views

The sum of the series $1+\frac{1\cdot 3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+\cdots$

The sum of the series $$1+\frac{1\cdot3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+ \cdots$$ is $\infty$, $1$, $2$, $4$. Can someone help me to solve this series.I am totally stuck on it.
3
votes
2answers
108 views

Sum of the series $\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\dots$

I am recently struck upon this question that asks to find the sum until infinite terms $$\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+.....∞$$ I tried my ...
3
votes
1answer
67 views

Show that $\lim_{n \to \infty}\prod_{k = n}^{2n}\dfrac{\pi}{2\tan^{-1}k} = 4^{1/\pi}$

MathWorld states that (see equation $(130)$) $$\lim_{n \to \infty}\prod_{k = n}^{2n}\dfrac{\pi}{2\tan^{-1}k} = 4^{1/\pi}$$ and attributes it to Gosper. I believe an approach to establish the formula ...
2
votes
1answer
28 views

Sequence, Monotone Convergence Theorem.

Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Use the Monotone Convergence Theorem to prove that either $x_n \rightarrow 2$ or $x_n \rightarrow 3$ as $n$ grows. ...
1
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1answer
156 views

True or False. Convergent subsequence

Is the statement true or false? If $(x_n)$ has a convergent subsequence,then $(x_n)$ is bounded. The statement is False. However, can someone please show me an example of a sequence with ...
0
votes
3answers
82 views

Sum of the infinite series $\frac16+\frac{5}{6\cdot 12} + \frac{5\cdot8}{6\cdot12\cdot18} + \dots$

We can find the sum of infinite geometric series but I am stuck on this problem. Find the sum of the following infinite series: $$\frac16+\frac{5}{6\cdot 12} + \frac{5\cdot8}{6\cdot12\cdot18} + ...
0
votes
3answers
32 views

sequence consisting of finite number of distinct values.

This is a question from my previous year analysis exam .The question says : Can we construct a sequence which converges but never attains its limits,such that its terms consist of a finite number ...
0
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0answers
29 views

Condition for a function $f: \mathbb R \rightarrow \mathbb R$ being right or left-continuous at $a \in \mathbb R$.

I know that $f: X \rightarrow \mathbb C$ is continuous if and only if for every convergent sequence $(x_n)$ in $X$ the identity holds $\lim_{n \rightarrow \infty} f(x_n) = f(\lim_{n \rightarrow ...
0
votes
1answer
24 views

Infinite summation of exponential $\sum_{n\in\mathbb{N}}e^{-n^k}$

For interger $k\geq 2$ is it possible to compute the sum and get an expression in terms of $k$? $\sum_{n\in\mathbb{N}}e^{-n^k}$
0
votes
2answers
15 views

Sequence converging to one.

Suppose that x_0 is a real number and x_n = [1+x_(n-1)]/2 for all natural n. Use the Monotone Convergence Theorem to prove x_n → 1 as n grows. Can someone please help me? I don't know what to assume ...
10
votes
0answers
316 views

Sorting of prime gaps

Let $g_n $ be the $n^{th}$ prime gap $p_{n+1}-p_n.$ If we re-arrange the sequence $ (g_n)$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
0
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2answers
24 views

Understanding and writing limit proofs

I got this question : Consider $a_n,\:b_{n\:}$ sequences such that for every n , $0\le a_n\le \:b_{n\:}$. Let $\lim _{n\to \infty }\left(\frac{b_n}{a_n}\right)\:=\:1$, and $a_n$ is a ...
0
votes
1answer
24 views

A geometric series weighted by a telescoping series

Is is possible to get a closed form expression of the sum of the series shown below: $$S_n = 1 + (a_1 - a_0)r + (a_2- a_1)r^2 + ... + (a_n - a_{n-1})r^n$$ where, $0 < a_0 < a_1 < ... < ...
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1answer
58 views

On the convergence of a series

Let $\{a_n\}$ be an increasing sequence of positive real numbers tending to $\infty$. Then $\sum\limits_{n = 1}^\infty \frac{a_n - a_{n - 1}}{a_n^\sigma}$ converges for any $\sigma >1$. Colud ...
1
vote
1answer
18 views

Decreasing sequence, bounded below.

Suppose that $0 \leq x_1 < 1$ and $x_{n+1} = 1 - \sqrt{1 - x_n}$ for all natural $n$. Prove that $x_n$ is decreasing and bounded below as $n$ converges. Attempt: Suppose that $0 \leq x_1 < 1$ ...
1
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1answer
27 views

Does the sequence converge or diverge?

I'm having trouble understanding how to do this one. If anyone could help I would be grateful. Does the sequence $$ \left\{ \sum_{n=1}^k \left( \frac{1}{\sqrt{k^2+n}} \right) ...
16
votes
4answers
527 views

Baffled by resolving number list

My son's Maths homework was to do with number patterns/sequences. "What is the nth term?". He'd done very well, but the last sequence was something like this: ...
16
votes
1answer
242 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
0
votes
1answer
21 views

Prove boundedness of recurrence relation

For a number sequence $\{y_n\}$ we know that $y_{n+1} = 2y_n-y^2_n$ If: $0<y_0<1$ show that $0<y_n<1$ for all integers $n>0$ I've tried solving the recurrence relation, but I couldn't ...
2
votes
1answer
56 views

Solving the recursion $y_n=2ny_{n-1}$ with Wolfram|Alpha

This is blowing my mind away ... this should be easy stuff! Starting with the recursive formula $y_n=2n*y_{n-1}$ where $y_1=5.$ I'm trying to come up with a formula for the series { 5, 20, 120, 960, ...
1
vote
1answer
57 views

Why do both trig functions have the same Macluarin series?

Both the degree version and the radian version of the trig functions have the same Maclaurin series, yet they are different. How is this possible? How can two different functions have the same ...
3
votes
2answers
107 views

Proving the bound $\left ( 1+\frac{x}{n} \right)^n \leqslant 3^x$, $\forall x \in \mathbb{R^+}$

I'm trying to directly prove the above bound. I have tried expanding it $$\left ( 1+\frac{x}{n} \right)^n = \sum_{k\geqslant 0} \binom{n}{k}\left ( \frac{x}{n}\right)^k$$ $$= \sum_{k=0\dots n ...
1
vote
0answers
17 views

Sequence terms being divisible

Here's a question I would like hints for: The sequence ${x_n}$ is defined by $x_{n+2}=6x_{n+1}-9x_{n}$ for $n \geq 0$ where $x_0=3$ and $x_1=18$. What is the smallest $k$ such that $x_k$ is ...
0
votes
1answer
30 views

Difficulty understanding Divergence Test

I'm studying Series and Diverge Test. But I'm having a problem understanding it. It says that, when the limit of it's partial sums is not equal to zero then it diverges. But then, there's also an ...
1
vote
1answer
14 views

Series expansions and perturbation

My professor said that $ f \left( y_1(x)+ \epsilon y_2(x)+... \right)= f(y_1(x)) +f'(y_1(x))\> (\epsilon y_2(x)+...) + ...$ but I have no idea how the series continues. Has anyone seen this ...
9
votes
4answers
200 views

Evaluating a series to order “three halves”

In doing some calculations related to one-dimensional Brownian Motion confined to a finite interval, I have come across functions such as $$ f(t) = \sum_{n=1}^\infty\frac{\exp(-n^2t)}{n^4}. $$ I ...
0
votes
0answers
79 views

Is it possible to determine the value of the following function?

Let $a_n$ and $b_n$ be a pair of generic sequences, and let $L$ be a constant. Consider the following function: $$f(x):=a_o+\sum_{n=1}^{\infty}\left({a_n}\cos{\frac{n\pi x}{L}+b_n\sin{\frac{n\pi ...
2
votes
3answers
53 views

Sum of Harmonic Numbers

Similar to this question , let $H_n$ be the $n^{th}$ harmonic number, $$ H_n = \sum_{i=1}^{n} \frac{1}{i}$$ Is there a similar method to calculate the following?: $$\sum_{i=1}^{n}iH_i$$