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

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3answers
34 views

How to prove $\sum\limits_{i=1}^{\infty}\dfrac{i}{2^i}$ converges?

What would be the simplest way to prove that $\sum\limits_{i=1}^{\infty}\dfrac{i}{2^i}$ converges?
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1answer
11 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
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1answer
16 views

Cauchy Sequences, not converging to zero

True or False? If $\{x_n\}$ and $\{y_n\}$ are Cauchy and $x_n + y_n > 0$, for all $n\in\mathbb{N}$, then $\left\{\frac{1}{(x_n + y_n)}\right\}$ cannot converge to zero. I believe the claim to be ...
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1answer
32 views

alternating series $\sum(-1)^na_n$ is divergent, then, is $\sum A_k$ divergent?

An alternating series $\sum\limits_{n=1}^\infty (-1)^na_n$ is divergent , $a_n\geq0$, and $\lim\limits_{n\to\infty}a_n=0$. Could we conclude that $\sum\limits_{k=1}^\infty A_k$ is divergent, too ? ...
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0answers
18 views

Alternating root in a sequence

Is there a way to have a closed form for $P_{n}$, where $P_{1}=x$ $P_{2}=\sqrt{x}$ $P_{3}=x$ $P_{4}=\sqrt{x}$ $\vdots$
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0answers
16 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
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1answer
50 views

Sum of series with cosines

I need to prove this: $$\sum\limits_{n=1}^\infty \frac{8}{(2n-1)^2 \pi ^ 2} \sin((2n -1) \frac{\pi x}{2}) \sin((2n -1) \frac{\pi z}{2}) = \min(x, z)$$ I got this: $$ \frac{8}{\pi ^ 2} ...
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1answer
31 views

Finding $N$ when sum is given

$1^2 + 2^2 + 3^2 + 4^2 + \cdots + N^2 = S$ Given $S$ How to find $N$. The Formula to Find $S$ from $N$ is: $S = \frac{N(N+1)(2N+1)}6$ so this gives me a cubic equation: $2N^3 + 3N^2 + N = 6S$ ...
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1answer
63 views

Plouffe's formula for $\pi$

Plouffe established the following formula for $\pi$ in $2006$ $$\pi = 72\sum_{n = 1}^{\infty}\frac{1}{n(e^{n\pi} - 1)} - 96\sum_{n = 1}^{\infty}\frac{1}{n(e^{2n\pi} - 1)} + 24\sum_{n = ...
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0answers
21 views

Sum of roots of terms of an Arithmetic Progression

Is there any easy way or formula to calculate sum of roots of an Arithmetic progression? For example if the arithmetic progression is $a+d$,$a+2d$,$a+3d$, $\ldots$, $a+nd$: How can I calculate ...
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1answer
61 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
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3answers
342 views

Example of a divergent sequence

I want to produce a divergent sequence for which $|x_n - x_{n-1}| \to 0$. So far, I've only been able to show that $$\frac{x_n}{n} \to 0$$, which doesn't really help.
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2answers
38 views

Show that $\sum_{k=0}^{+\infty}ka_k=\sum_{k=0}^{+\infty}\sum_{i=k+1}^{+\infty}a_i$.

Let $(a_k)_{k\in\Bbb{N}}$ a family of positive real. Show that $$\sum_{k=0}^{+\infty}ka_k=\sum_{k=0}^{+\infty}\sum_{i=k+1}^{+\infty}a_i$$ So far, I have : By definition ...
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4answers
26 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 ...
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1answer
36 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 ...
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1answer
28 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. ...
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1answer
26 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 ...
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1answer
34 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 ...
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2answers
83 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} ...
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1answer
24 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: ...
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2answers
24 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, ...
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3answers
54 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 ...
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3answers
100 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?
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2answers
97 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
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3answers
35 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 ...
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1answer
49 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 ...
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5answers
111 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 ...
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0answers
20 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: ...
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1answer
157 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 ...
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0answers
31 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 ...
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1answer
27 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 $k$? $\sum_{n\in\mathbb{N}}e^{-n^k}$
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2answers
111 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 ...
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3answers
33 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 ...
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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 ...
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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} + ...
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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
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. ...
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1answer
19 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 $0 \leq x_1 < 1$ ...
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1answer
29 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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
15 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 ...
3
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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 ...
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0answers
84 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 ...
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1answer
35 views

Geometrical Sequence

I have a sequence $U(n)$ with $U(n+1) = f(U(n))$ with $f(x) = ax+b$. I'm supposed to express $U(n)$ as a function dependant on $n$. Doing so with an auxiliary sequence $V(n) = U(n) - ɑ$. Where $ɑ$ ...
1
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1answer
27 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
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3answers
36 views

Common ratio of a GP

Question: If $f$ is a function satisfying $f(x + y) = f(x)f(y)$ for all $x,y$ that are natural numbers; such that $f(1) = 3$ and $$\sum^{n}_{x=1}f(x) = 120$$ find the value of n. I don't understand ...
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0answers
48 views

Convergence of the series $\sum_{n=1}^\infty (1+\frac{1}{\sqrt{n}})^{-n^\frac{3}{2}}$

Test the convergency of the series $$\sum_{n=1}^\infty \left(1+\frac{1}{\sqrt{n}}\right)^{-n^\frac{3}{2}}.$$ We know that, if $\sum_{n=1}^\infty U_n$ is convergent, then ...