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

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0answers
4 views

Proving that Changing the Index of the Lower Bound of a Convergent Infinite Series Does Not Affect the Convergence

Would someone help me in proving that the following theorem is true? Let $j$ be a positive integer. Show that $$\sum\limits_{k=0}^\infty a_{k} \quad\textrm{ converges iff }\quad ...
0
votes
0answers
14 views

Values of $x$ for which $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges.

For what values of $x$ the infinite series $\sum [(n^3+1)^\frac{1}{3}-n]x^n$ converges?
1
vote
2answers
29 views

Harmonic progression based question

If $a,b,c,d$ are distinct positive numbers in harmonic progression, then (A) $a+b>c+d$ (B) $a+c>b+d$ (C) $a+d>b+c$ (D) none of these I tried ...
1
vote
3answers
32 views

Calculating limit-sequences

So i have this limit to calculate: $\lim_{n\to\infty}\frac{[x] + [2x]+ ... +[nx]}{n^2}$ And i tried to make some boundaries and got this two limits: $\lim_{n\to\infty}\frac{[x] + [x]+ ... ...
0
votes
4answers
64 views

Calculate the sum of this series

$$ \sum_{n=1}^\infty \frac{1}{n^2 3^n} $$ I tried to use the regular way to calculate the sum of a power series $(x=1/3)$ to solve it but in the end I get to an integral I can't calculate. Thanks
0
votes
1answer
27 views

What is the a[n] for this sequence and its ∑ a[n] (k=1..12)

What is the $a_{n}$ for this sequence? $$\left \{333, 2\cdot 333+0.1\cdot \frac{333}{12}, 3\cdot 333+0.1\cdot2\cdot \frac{333}{12},4\cdot333+0.1\cdot3\cdot \frac{333}{12},\dots, a_{n}\right \}.$$ And ...
5
votes
3answers
46 views

Finding the general formula for $a_{n+1}=2^n a_n +4$, where $a_1=1$.

Problem: Find the general formula for $a_{n+1}=2^n a_n +4$, where $a_1=1$. Find the sum of its first $2n$ terms with odd subscript. My effort: It seems to me that $a_{n+1} / ...
1
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3answers
37 views

An infinite sum of lengths.

Let both the base and height of the following triangle have the length 6m. If the lines drawn inside the triangle bisect each right angle formed while proceeding. Then find the length of all these ...
0
votes
0answers
59 views

Is 1 really equal to 0.99999999… [duplicate]

I heard a few times that 1 is equal to $0.9999 \dots$ (infinite nines). I know that the limit of this is actually 1, but does that that the equivalency hold here? Can't we argue that $1 - 0.99999 ...
-2
votes
1answer
38 views

Convergence of $\sum_{n=1}^{\infty}\frac{(n!)^n}{(n^n)^2}$ [on hold]

Test the convergence of the following series $$\sum_{n=1}^{\infty}\frac{(n!)^n}{(n^n)^2}$$
12
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1answer
53 views

Rearrangements that never change the value of a sum

Which bijections $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ have the property that for every sequence $\{a_n\}_{n=1}^\infty$, $$ \lim_{n\to\infty} \sum_{k=1}^n a_n = \lim_{n\to\infty} \sum_{k=1}^n ...
4
votes
2answers
98 views

Weird thing about $\sum_{k=2}^{\infty}(-1)^k\zeta{(k)}$

Consider the sum $S=\sum_{k=2}^{\infty}(-1)^k\zeta{(k)}$. By a simple manipulation, we can show: $$ ...
0
votes
0answers
37 views

Need help solving min, max, inf and sup of sequence!

We've been given the sequence $x_n=(-1)^n \cdot \frac{\sqrt{n}}{n+1}+\sin \frac{n \pi}{2}$. I have to find the min, max, inf and sup (if they exist), and also find the points of accumulation. Any ...
6
votes
1answer
63 views

Closed form of series involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number and $\phi$ be the golden ratio, that $\phi = \frac{1+\sqrt{5}}{2}$. Find a closed form for the sum: $$\sum_{n=0}^{\infty} \frac{1}{(5\phi)^n(n+2)} ...
1
vote
3answers
44 views

What does it mean for a function to “preserve the limits of sequences”?

I've translated the English Wikipedia page "Limit of a sequence". What does the following statement mean? In fact, any real-valued function f is continuous if and only if it preserves the limits ...
1
vote
1answer
85 views

Convergence of $\sum \frac{2n+1}{(n^2+n)^n}$

I have to choose the right option: The series $$\sum_{n\geq 1} \frac{2n+1}{(n^2+n)^n}$$ a. Converges to 1. b. Converges to a number >1. Using ...
1
vote
2answers
56 views

Trouble with finding the limit of this sequence

Well I was trying to find the limit of - $$ \lim_{x\rightarrow \infty } \lim_{n\rightarrow \infty} \sum_{r=1}^{n} \frac{\left [ r^2(\sin x)^x \right ]}{n^3}$$ obviously $$ ...
-1
votes
0answers
24 views

infinite limit of factorial funtion [on hold]

What will be the limiting value of $$\lim_{n \to \infty} \, \sum_{j=0}^{\left[\frac{n}{c}\right]}\frac{(1+a)_{bn}}{(n-cj)!}$$ where $a,b,c \in \mathbb{N}$ and $(x)_{m}$ is the Pochhammer symbol ?
10
votes
2answers
120 views

$\lim x_n^{x_n}=4$ prove that $\lim x_n=2$ [duplicate]

Let $(x_n)$ be a sequence of real numbers, such that: $\lim x_n^{x_n}=4$, prove that $\lim x_n=2$ I'm not sure if my proof is right. I assumed that $\lim x_n $ isn't 2 and using Cauchy's criterion: ...
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0answers
45 views

Julia and Mandelbrot Sets [on hold]

I need to know how escape, prisoner, Julia and Mandelbrot sets work. Are they all in one sequence or are they separate.
3
votes
1answer
50 views

Prove or disprove that a series is convergent

I was given the following task which I struggle with. Prove the following statement, or disprove it by giving a counter example: if $\sum_{n=1}^\infty a_n$ is convergent then $\sum_{n=1}^\infty ...
2
votes
2answers
81 views

Convergence of Sequence

If I know that the sequence $\{a_n\}$ converges to $a$, then to prove that the sequence $\{ca_n\}$ (for a constant $c$) converges to $ca$, I would basically want $|ca_n - ca| < \epsilon$... So, to ...
1
vote
0answers
33 views

Can someone suggest some reference for a particular kind of infinite series

Can someone suggest some reference for the properties of the series $\lim_{n \to \infty} \sum_{i = 0}^{n} f(n, i)$ For example, when can we apply approximations to $f(n, i)$ that only works when $n ...
1
vote
1answer
20 views

A question about fixed-point iteration sequence of a two times continuously differentiable function

I am stuck at this problem: Let $g:[a,b]\to[a,b]$ be a 2 times continuously differentiable function that satisfy: for all $x\in [a,b]$, $g''(x)\neq 0$ And let $s$ be an arbitrary fixed-point of ...
2
votes
3answers
67 views

Why would the cubic have $5$ roots?

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$? $P(x) = ...
6
votes
1answer
116 views

Find the sum of the series below

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ What I have tried... We are looking for ...
1
vote
1answer
74 views

Convergence to $0$ of a certain series.

I was wondering whether or not the following holds - I didn't manage to get anywhere using standard tricks from elementary analysis. ...
2
votes
2answers
72 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
1
vote
1answer
47 views

Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
5
votes
1answer
158 views

Can we find this limit?

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence such that $a_1=1$ and $a_{n+1}=\sqrt{a_n^2+a_n}$ for $n\geq 0$. Is it possible to find $\displaystyle\lim_{n\to\infty}\frac{a_n}{n}$ ? I have no any idea. ...
1
vote
3answers
76 views

Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$ [on hold]

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
0
votes
0answers
47 views

Is it possible to find the exact value of this

Is it possible to find the exact value of the infinite series ? $$\sum_{n=1}^\infty \frac{2^n}{(1+\sqrt{2})^n+1}$$ I have no any idea. Thank you for helping.
1
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0answers
13 views

Determining an integer to approximate error using the Alternating Series Test

I need to determine the smallest value m, such that: $$ \left |\int_0^{0.1} arctan(x^2) dx - \sum_{n=0}^m (-1)^n \frac{(0.1)^{(4n+3)}}{(4n+3)(2n+1)} \right| < 10^{-8} $$ Using the Alternate ...
0
votes
0answers
30 views

Proof that the sum of a certain infinite series can be bounded to zero

$\forall 0 < \alpha < 1$, there exists $\lambda > 0$, $k > 0$, s.t. $$ \lim_{n \to \infty} \sum_{w = 1}^{\lambda n} \binom{n}{w} \frac{1}{2^{\alpha n}}\left(1 +\left(1 - ...
0
votes
1answer
21 views

Summation of finite power seires

Is it possible to find a close form solution for $S_1$. $S_1$ is defined as follows: $S_1=\sum_{k=b}^{\infty}\frac{x^k}{k!}$ ; Where $0<x<b<\infty$ If $b=0$ then $S_2 = e^x$. But how do we ...
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votes
0answers
17 views

Cauchy second theorem [on hold]

Where to use Cauchy second theorem I mean in it can be used only in the case of powers and factorials or somewhere else also.
1
vote
1answer
37 views

Tough problem on sum of infinite series [on hold]

I've been working on the problem for quite a while but have no idea how to approach it. This proposition arises from a practical probabilistic bound problem, but it seems very deep. Lots of thanks to ...
2
votes
4answers
66 views

Does $\displaystyle\sum^{\infty}_{n=1}\left(\frac{n!}{n^n}\right)$ converge or diverge? [duplicate]

Does $\displaystyle\sum^{\infty}_{n=1}\left(\frac{n!}{n^n}\right)$ converge or diverge? I've tried the ratio test, but i'm unsure if I can continue this way. ...
0
votes
1answer
51 views

Is that form of Cesàro's theorem correct?

First, for sequences, we know that : "If a sequence $(a_n)_{n \ge 1}$ converges to $l\in \mathbb{R}$, then the sequence $(b_n)_{n \ge 1}$ defined by : $b_n=\frac{1}{n} \sum \limits_{k=1}^{n}a_k$ ...
3
votes
0answers
80 views

Is there a closed-form approximation to a band-limited sawtooth?

A partial Fourier Series with no coefficients is equal to the closed form expression: $${A \over n} \sum_{k=1}^n \cos(k\theta) = {A \over 2n} \left\{{\sin([2n + 1]\theta/2) \over \sin(\theta/2)} - ...
2
votes
4answers
61 views

Superior limit of a certain sequence

Let $(x_n)$ be the sequence: $$\{1, 2, 1 + \frac12, 2 + \frac12, 1 + \frac13, 2 + \frac13, \ldots \}$$ I (think I) understand why $\lim \inf x_n = 1$, but I'm not sure why $\lim \sup x_n =2$. Any ...
1
vote
2answers
24 views

Can we replace the limit of a sequence with that of a function?

Let $f$ be a function defined in $[1,\infty]$. If $\lim_{x\to\infty}f(x) = L$ and $a_n = f(n)$ for integer $n\ge 1$ then $\lim_{n\to\infty}a_n = L$. Found this theorem in many references, but ...
2
votes
2answers
83 views

Find the limit of an infinite series

My intuition was to try and see if the series is a Riemann Sum of a function and then see what happens but I can't really see which function fits here. Thanks!
0
votes
3answers
52 views

What is the limit of this sequence as n->infinity? [on hold]

Find the limit of the following sequence $n^{\ln(n)/n}$ as $n\to\infty$? Please answer without using L'Hopital
0
votes
5answers
112 views

Find limit of the following sequence?

Find the limit of $\frac{\log(n+1)}{\log(n)}$ where $n\rightarrow\infty$. Here $n$ is a natural number so I guess we can't use L'Hopital
9
votes
2answers
180 views

Proving a sequence converges when combinations of consecutive terms converge

Problem: Let $\{x_n\}$ be a sequence of real numbers such that $$\lim_{n\to\infty} 2x_{n+1}-x_n=L \in \mathbf{R}.$$ Prove that $x_n \to L$ as $n\to\infty$. I can see that if $\{x_n\}$ converges to a ...
2
votes
3answers
39 views

calculate two-fold difference

These are a series of numbers that increase two folds: $$0.125, 0.25, 0.5, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024$$ If I pick up two numbers, say $0.5$ and $128$, I want to know know how may ...
2
votes
1answer
46 views

Geometric series and complex numbers [duplicate]

I'm new to this site, english is not my mother tongue, and I'm just learning LaTeX. I'm basically a noob, so please be indulgent if I break any rule or habits. I'm stuck at proving the following ...
7
votes
1answer
127 views

Proving convergence for a series.

Let there be given that $\displaystyle\sum_{n=1}^{\infty}a_{n}$ is a convergent series and $a_{i}$ is not negative for every $i\in\mathbb{N}$. And i want to prove that this series is also convergent ...
-1
votes
4answers
76 views

The sum of the series $\sum_{n=1}^{\infty}\sin^n(k)$

What is $$\sum\limits_{n=1}^{\infty}\sin^n(k)?$$ Can you find what is the sum of that series. It is convergent not divergent. What if $k=\frac{\pi}{6}$?