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

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1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
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1answer
29 views

Writing dense sets in terms of set of integers

Can we write every dense set in $\mathbb R$ as {$x_n$}$\mathbb Z$ , where {$x_n$} is a real sequence with limit $0$ ?
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0answers
53 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
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0answers
62 views

sum of reciprocal power-1

I found this in my old notebook $$\sum_{n \text{ perfect power}} {\frac{1}{n-1}} = 1$$ and this was my "proof" $$ \begin{align} \frac{1}{1}+\frac{1}{2}+\cdots ...
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2answers
31 views

Divergence of a recursive sequence

If $(x_n)$ isthe sequence defined by $x_1=\frac{1}{2}$ and $x_{n+1}=\sqrt{x_n^2 +x_n +1}$, show that $\lim x_n = \infty$ Ive tried a couple of things but none of them helped. Ive tried to suppose, by ...
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2answers
156 views

Prime Number Sum Sequence (Amateur)

SOLVED: This is false Beginning with 3, add the next consecutive prime (2) and then take that sum (5) and add that to next consecutive prime (3) to get (8), and so on... $$ 3 + 2 = 5 $$ $$ 5 + 3 = 8 ...
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1answer
25 views

Count of paths between N points

I am trying to arrive at a formula that will give me the number of distinct paths between a set of discrete points on a map. I have worked out that I can calculate it using a series of additions: ...
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1answer
27 views

Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?

I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of ...
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2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
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5answers
57 views

Arithmetic Progression.

Q. The ratio between the sum of $n$ terms of two A.P's is $3n+8:7n+15$. Find the ratio between their $12$th term. My method: Given: $\frac{S_n}{s_n}=\frac{3n+8}{7n+15}$ ...
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1answer
58 views

What is the closed form for this sequence, powers of $4$?

What is the closed form for this sequence: 1, 4, 12, 40, 148, 576, 2284, 9112, 36420, 145648, 582556, 2330184, 9320692, 37282720, 149130828, 596523256, 2386092964, 9544371792, ...
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3answers
59 views

How to show that $a_{n+1} = \frac{a_n^2 +3}{4} $ is increasing.

I'm not good at finding whether a sequence is increasing or decreasing. $a_{n+1} = \dfrac{a_n^2 +3}{4}$ is the recursive sequence where $ a_1 =0$ How to get the approach to do something like this? ...
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0answers
191 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 ...
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0answers
25 views

A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit

In one of my books there was an exercise to prove that: A sequence $a_n$ converges to $+\infty$ iff it has $+\infty$ as it's only partial limit (I know a sequence doesn't really converge to ...
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1answer
45 views

Show that C is a closed convex subset and its element of minimum norm

I have a lot of problems with the following exercise that I can't solve. Let $(L^1((0,1)), \|\cdot\|_{L^1}):=(E,\|\cdot\|)$ and $$C:=\{u\in E:u\geq0 \text{ a.e } x\in (0,1),\quad T(u)\geq 1\},$$ ...
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1answer
77 views

How prove $T_{n}\neq 0$,if $T_{n+2}=(1-2c)T_{n+1}+(2c+a-c^2)T_{n}-(a-c^2)T_{n-1}$

Question: Assmue that the postive integer $a,c$ ,such $\lfloor \sqrt{a} \rfloor=c$ ,Now let sequence $$y_{1}=1,y_{2}=-2c, y_{n+2}=-2c\cdot y_{n+1}+(a-c^2)y_{n},n\ge 1$$ show that ...
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3answers
100 views

Show that $q^4+2pq^2 +p^2 = 2pq -(pq)^2 -1$ becomes $p^3+q^3+3pq-1=0$.

I know that these two are exactly the same equation but I can't seem to prove it. You are also given that $p+q=1$. This is a follow up from a similar question.
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0answers
50 views

Analytic solution for system of Digamma function equations

Peace be upon you, There is a while, I am seeking for the analytic solution of $\alpha$ and $\beta$ in this system \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ ...
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1answer
64 views

By plugging $p=1-q$, into the $3$ equations show that $x=y=z$

By plugging $p=1-q$, into the 3 equations: $$\begin{cases} z=py+qx \\ x=pz+qy \\ y=px+qz \end{cases}$$ show that $\boxed{x=y=z}$ This is from the final part of question 7 in this STEP paper, and is ...
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6answers
215 views

If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_n $, prove that $\sum \frac{a_n}{b_n}$ diverges

Let $\displaystyle \sum a_n$ be convergent series of positive terms and set $\displaystyle b_n=\sum_{k=n}^{\infty}a_n$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges. I could ...
2
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3answers
95 views

How to prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier Series [duplicate]

Can we prove that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ without using Fourier series?
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2answers
63 views

Prove $A=\{x\in (1,2): \text{the decimal expansion of $x$ contains only } 1,3 \text{ or }5\}$ is compact

Prove $A=\{x\in (1,2): \text{the decimal expansion of $x$ contains only}~ 1,~3~\text{or}~5\}$ is compact It is bounded being a subset of $(1,2)$ the only thing left to prove is that it is closed. ...
2
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4answers
48 views

Show that a sequence is bounded if and only if there exists a K $\in\mathbb{R}$ such that $|\ a_n\ | \leq K$ $\forall n\in \mathbb{N}$.

Show that a sequence is bounded if and only if there exists a K $\in\mathbb{R}$ such that $|\ a_n\ | \leq K$ $\forall n\in \mathbb{N}$. Is it correct/enough for me to show the following: Let ...
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2answers
23 views

Closing the gap between convergence and divergence of $\sum_n 1/n^{1 + 1/f(n)}$ for increasing $f$.

So I have managed to show that $\sum_n 1/n^{1 + 1 / \log n}$ diverges and $\sum_n 1/n^{1 + 1 / \log \log n}$ converges. But the growth rates of $\log n$ and $\log \log n$ are very different. Can ...
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5answers
41 views

Does $\sum_{i=1}^\infty \frac{1}{(a+i)^b}$ converge for $b>1$?

Does $$\sum_{i=1}^\infty \frac{1}{(a+i)^b}$$ converge for $b>1$? What is the name of this series?
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0answers
24 views

Convergence of Newton series

What is the condition for a real valued function of a real variable to have a Newton series which converges to that function pointwise? It feels like there should be a condition similar to that for ...
3
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1answer
49 views

Limit of a sequence of averaged numbers?

Let $a_0 = 0$, $a_1 = 1$, and $a_n = \frac{a_{n-1}+a_{n-2}}{2}$ for all $n \ge 2$. Consider $\lim \limits_{n \to \infty} a_n$. Using a quick python script I found that for large $n$ $a_n$ tends to ...
0
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1answer
22 views

Recurrence relation with geometric sequence

Reading the seemingly excellent book Basic Stochastic Processes (Brzezniak), but got confused by a derivation, page 86-87 specifically. We arrive at this formula: $x_{n+1} - \frac{q}{p+q} = ...
2
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3answers
99 views

Double summation index problem

I often meet the following situation: $$\sum\limits_{n=0} ^\infty \sum\limits_{k=0} ^n f(k)g(n-k)=\sum\limits_{p=0} ^\infty \sum\limits_{q=0}^\infty f(p)g(q)$$ While intuitively this is very clear ...
0
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1answer
58 views

Series problem:

Parts i) & ii) I can solve. For part iii) I get $z=py+qx$ [For $n=0$] $x=pz+qy$ [For $n=1$] $y=px+qz$ [For $n=2$] leading to $(1-pq)x=(q^2+p)z$ [1] $(1-pq)z=(q^2+p)y$ [2] ...
2
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1answer
63 views

If $ \sum a_n$ diverges and $\lambda_n \to \infty$, does the series $ \sum \lambda_na_n$ diverge?

Suppose that the series $\displaystyle \sum a_n$ diverges and $\lambda_n \to \infty$. Does the series $\displaystyle \sum \lambda_na_n$ diverge? And what happens if $\{\lambda_n\}$ is an unbounded ...
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2answers
35 views

What does this sequence converge to?

Could you please help me solve this question relating to sequences? Suppose that a sequence $\{a_n\}$ converges to $\pi$. Then the sequence $\{\cos(a_n)\}$ _____. The answer is converges to $-1$ ...
5
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5answers
257 views

Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$

$T(0)=1 \\ T(n) = 1 + \sum_{j=0}^{n-1}T(j) \\ $ Show that $T(n) = 2^n$. I know how to prove this by induction, but I would like to know how to show this using first principles. Edit: The way I want ...
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2answers
38 views

For the summation $\sum\limits_{n=0}^\infty\frac{(-1)^{n+1}n!}{1*3*5*…(2n+1)}$ when performing the Ratio test, why is the $(-1)^{n+1}$ term removed?

For the summation $\sum\limits_{n=0}^\infty\frac{(-1)^{n+1}n!}{1*3*5*...(2n+1)}$ when performing the Ratio test, why is the $(-1)^{n+1}$ term removed (this is what CalcChat shows). I understand that ...
0
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1answer
45 views

Are there other rational/algebraic term series for $e$, and do any converge faster or are faster to compute than the classical series?

We have the classical rational term series $e = \sum_{n=0}^\infty 1/n!$ which converges incredibly fast. But are there other series for $e$ that have all rational or algebraic terms with closed form ...
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1answer
22 views

Formula of Squaring Sums / Integrals

I'm trying to find a proof for the identities (which I use quite often) $$\left ( \int_{a}^{\infty}f(x)\,dx \right )^2=\int_{a}^{\infty}\int_{a}^{\infty}f(x, y)\,dx\,dy$$ and similarly for the series ...
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2answers
17 views

Find the common ratio, if the sum of first 8 terms in GP(geometric progression) is 5 times the sum of first 4 terms also in GP.

I tried it by supposing, (the supposition where “if the number of terms is odd, we take the middle term as ‘a’ and the common ratio as ‘r’. If the number of terms is even, we take ‘a\r’ and ‘ar’ as ...
2
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2answers
54 views

Verifaction of convergence/divergence exercise

I have the following assignment in my textbok: Series $\sum_{n=0}^{\infty}c_{n}3^n$ is convergent. Based on that can we conclude that the following series coverge: a) $\sum_{n=0}^{\infty}c_{n}2^n$ ...
6
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1answer
353 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
10
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3answers
193 views

Why is $\sum_{n=-\infty}^{\infty}\exp(-(x+n)^2)$ “almost” constant?

I did some numerical approximation of $$\sum_{n=-\infty}^\infty \exp(-(x+n)^2)$$ and found that this function is "almost" constant ($\approx 1.772$). Why does the sum fluctuate little? Is there a ...
2
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1answer
96 views

Generalised Fibonacci

How to find the nth term of the recurrence in $\log n$ time. $$ \begin{array}{rcl} F[n]&=&F[n-1]+F[n-3]\\ F[2]&=&1\\ F[3]&=&2\\ F[4]&=&3\\ F[5]&=&4 \end{array} ...
4
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1answer
64 views

Limit of differences of truncated series and integrals give Euler-gamma, zeta and logs. Why?

In the MSE-question in a comment to an naswer Michael Hardy brought up the following well known limit- expression for the Euler-gamma $$ \lim_{n \to \infty} \left(\sum_{k=1}^n \frac 1k\right) - ...
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1answer
54 views

What pattern rules this sequence? [closed]

$$ 1, 85, 68257, 4585645, 73773171109 $$ I have tried Wolfram Alpha and OEIS but both failed me. Anything helps. Thanks!
1
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1answer
26 views

Summation of series- substitution

If we have $\sum_{n=0}^{\infty}nf(n)=C, C\ne0\tag 1$, C is a constant, can we find a closed form for f(n)?. NB : Given condition is that $\sum_{n=0}^{\infty}f(n)$ converges to a constant value $K$ ...
0
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1answer
49 views

Sequence in $\mathbb R^2$ converges if and only if it is Cauchy

How to prove that a sequence $(x_n)$ in $\mathbb R^2$ converges if and only if it is Cauchy? I've proven that the triangle inequality holds for the euclidean norm of vectors.
2
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1answer
35 views

Uniform convergence of $xe^{-nx}$

Does the sequence $(f_n)$ on $[0, \infty)$ given by $ f_n(x) = > xe^{-nx} $ converge uniformly? This is from Bartle's Elements of Real Analysis. I've already proven that the sequence is ...
0
votes
2answers
65 views

Sequential Algebraic Problem:

The first part i) I can do. For part ii) this is how far I can get: If n is odd then $y=px+qy$ If n is even then $x=py+qx$ After some rearranging i end up with $y(1-q)=px$ & $x(1-q)=py$ and ...
2
votes
2answers
59 views

Find the first 5 coefficients of the series $\frac{6x}{x+9} = \sum_{n=0}^\infty C_n x^n$

I rewrote the equation series as $$ \frac69 \sum_{n=0}^\infty \left(\frac{-1}{9}\right)^n x^{n+1} $$ And therefore have coefficients of $C_0 = 6/9, C_1 = \left( 6/9 \right) ...
1
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3answers
143 views

How do you derive the continuous analog of the discrete sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …$?

I was wondering what the rate of growth of the sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...$$ was, and found the related question, $n$th term of $1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5...$, in which one of ...
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2answers
98 views

What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...