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

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-3
votes
0answers
44 views

$\sum_{n=0}^{\inf}n=- \frac{1}{12}$ Is it real? Why? [duplicate]

I assume many of you have stumbled upon the widely spread video that "proofs" the statement in my question. That video contains some serious mistakes (bracketing divergent series etc.), but the result ...
-1
votes
1answer
45 views

Intersection of a line with a curve given as a geometric series [closed]

I need to calculate the point at which this curve intersects with the line $3y$$=$$x$$-$$1$. I understand that I need to substitute the curve and line to form one equation, but am unsure of how to ...
-3
votes
0answers
27 views

help with showing a series is divergent [closed]

I tried unsuccessfully to show by convergence tests that the series $$\sum_{n=1}^\infty{\ln^nn\over n^2}$$ is divergent , cant seem to find a way. help would be very appreciated , thanks in ...
1
vote
4answers
177 views

Proof the golden ratio with the limit of Fibonacci sequence [duplicate]

Let $F_n=F_{n-1}+F_{n-2}$ the Fibonacci numbers, and $\phi=\frac{1+\sqrt5}{2}$ The exercise asks me to prove that: $\lim\limits_{n \to \infty}\frac{F_{n+1}}{F_n}=\phi$. Sorry as can be proceed??
2
votes
2answers
20 views

Proving two Recurrence Relations

I encountered this problem in the International baccalaureate Higher Level math book, and my teacher could not help. If anyone could please take the time to look at it, that would be great. I need ...
0
votes
1answer
28 views

Convergence of $\sum_{n=0}^\infty \frac{\sqrt{1+x^n}}{x^n}$ in the case $x<0$ and an analogous problem with $\sum_{n=0}^\infty \frac{x^n}{2+x^n}$

Let $$\sum_{n=0}^\infty \frac{\sqrt{1+x^n}}{x^n}.$$ My first question is: for what values of $x$ is this series possible? I can only say that it is not defined for $x = 0$, but are there other ...
5
votes
1answer
48 views

Why is the Cauchy product of two convergent (but not absolutely) series either convergent or indeterminate (but does not converge to infinity)?

It is well-known that the Cauchy product of two absolutely convergent series is absolutely convergent. However, my professor added (without giving a proof) that if the series are convergent ...
-6
votes
0answers
32 views

Help and solve for me on Sequences [closed]

solve for me this sequence. The number of the sequence is 17, if the second number is 3, what is the first number in that sequence
0
votes
0answers
25 views

Algorithm for generating a series of contrasting colours

On a computer screen, colours can be defined as having 0-255 units of red, green and blue. This creates a 3-dimensional colour space with $256^3$ different colours, from 0-0-0 for black to 255-255-255 ...
-2
votes
0answers
32 views

$\textrm{lim sup}$ of a function [closed]

I need to compute a limit: Given a series with the sum equal to $1$, I need to compute $\textrm{lim sup}\ (A_n)^{1/n}$. Since the sum is $1$, I assume that the value $A_n$ which actually should tend ...
9
votes
2answers
4k views

How do I prove this sum is not an integer

Assume that $k,n\in\mathbb{Z}^+$. Prove that the sum \begin{equation*} \dfrac{1}{k+1}+\dfrac{1}{k+2}+\dfrac{1}{k+3}+\ldots +\dfrac{1}{k+n-1}+\dfrac{1}{k+n} \end{equation*} is not an integer. The ...
2
votes
3answers
77 views

Does $\sum_{n=2}^\infty\frac{1}{n\ln n!}$ converge?

Let $$\sum_{n=2}^\infty \frac{1}{n\ln n!}.$$ It is equal to $$\sum_{n=2}^\infty \frac{1}{n(\ln n + \ln(n-1) +...+ ln(2))}.$$ But now what should I do to prove that it converges? (I have tried root ...
0
votes
0answers
20 views

Algorithm for filling in points around a circle with increasing density

The aim of this question is to decide on the order in which to download a series of high-resolution files that together represent a 720° rotation around an animating object. When all the files are ...
9
votes
6answers
311 views

Definition of a geometric sequence

Is the sequence $0, 0, 0, 0 ...$ geometric? If so how would you define it? In order to define a geometric sequence you need the first term, and the ratio of terms. In this case you could have: $a = ...
0
votes
2answers
33 views

Study: $\sum_{n=1}^\infty (\sin(\sin n))^n$, $\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$, and $\sum_{n=1}^\infty \frac{ \sin (x^n)}{(1+x)^n} $

Let $x \in \mathbb{R}$. I have to study the convergence of the following three series: $$\sum_{n=1}^\infty (\sin(\sin n))^n$$ $$\sum_{n=1}^\infty \frac{n \sin (x^n)}{n + x^{2n}}$$ ...
4
votes
2answers
4k views

How do you solve this series question? : $\cos (n\pi )/ \ln(6n) $

The problem is, Select the FIRST correct reason on the list why the given series converges. A. Geometric series B. Comparison with a convergent p series C. Integral test D. Ratio test E. ...
1
vote
2answers
28 views

Series definied by recurrence relation

Define a sequence recursively by $$\large{a_1 = \sqrt{2},a_{n+1} = \sqrt{2 + a_n}}$$ (a) By induction or otherwise, show that the sequence is increasing and bounded above by 3. Apply the monotonic ...
0
votes
1answer
25 views

How to show this is decreasing

I'd like to show $$\sum_{i=1}^n \frac{1}{i((n+1)-i)} $$ is decreasing for n>1, which is Cauchy product of $$\sum_{i=1}^n \frac{1}{i}$$ Numerical computation until n=50 shows it's decreasing but I ...
0
votes
1answer
34 views

Calculate the sum of the series $\sum_{n=0}^{\infty} \frac{1}{a_{n}\cdot a_{n+1}\cdot\ldots\cdot a_{n+7}}$, where $a_k = ak + b$

Let $a_k = ak + b$; define the following series: $$\sum_{n=0}^{\infty} \frac{1}{a_{n}\cdot a_{n+1}\cdot\ldots\cdot a_{n+7}}.$$ I have to prove that this series converges and I have to find its sum. ...
0
votes
1answer
1k views

Find an expression for the area under the graph of f(x) as a limit?

$f(x) = \frac{2x}{x^2 +1}, 1 \leq x \leq 3$ Basically, I need to find an expression for the area under the graph within these intervals for the function as a limit. I understand the concept of the ...
6
votes
1answer
69 views

A limit with the harmonic series

How can we prove the following (similar) limits? $$\sum_{k=1}^n \frac{1}{k} (\ln 2 - \frac{1}{n+2} - \frac{1}{n+3} - \cdots -\frac{1}{2n + 2}) \to 0. $$ $$\sum_{k=1}^n \frac{1}{k} (\ln 3 - ...
6
votes
2answers
243 views

Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$

I want to show that $$\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})} = \frac{n^2-1}6$$ With induction I don't know how I could come back from $\frac{1}{1-\cos(\frac{2k\pi}{n+1})}$ to ...
2
votes
0answers
15 views

Upper bounding Lerch zeta function

Let $\Phi\left(z,s,a\right) $ be a Lerch Trascendent. $$\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}}{1-ze^{-t}}dt.$$ Can we upper bound the above in ...
1
vote
0answers
23 views

Cauchy Criteria for Series

We know that the Cauchy Criterion of a series is as follow: Theorem: A series $\sum\limits_{i=1}^{\infty}x_i$ converges iff for all $\epsilon>0$ there is an $N\in \mathbb{N}$ so that for all ...
2
votes
3answers
35 views

Number sequence - arithmetic sequence difference constant - find formula

I searched a way to find out a formula to predict the nth number of a given sequence, but I did not find a way matching my case. Arithmetic sequence: I read that a good way is to find the constant ...
0
votes
2answers
35 views

Find All Positive Pairs of $(\alpha,\beta)$ Such that $\,\lim_{n \rightarrow \infty}\frac{\sum_{k=1}^{n}a_k}{n^{\alpha}}=\beta$

Consider the following sequence $(a_n)=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,...)$ Find all pairs of $(\alpha,\beta)$ of positive number such that $$\lim_{n \rightarrow ...
2
votes
5answers
92 views

To evaluate $\lim_{n\to\infty} \dfrac{10^n}{n!} .$

I am having a lot of trouble evaluating $$\lim_{n\to\infty} \dfrac{10^n}{n!} .$$ I know intuitively that $n!$ is much larger and this expression should go to zero but I just can't figure out how to ...
118
votes
15answers
8k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
4
votes
1answer
38 views

Order of Growth of a Sum

Let $n>k$ and $r$ be arbitrary positive integers. Define $q=k/n$. I want to show that $$ \sum_{i=0}^{rk} \binom{rn}{i}q^i(1-q)^{rn-i}(rk-i)=\Theta(\sqrt{r}) $$ as $r\rightarrow \infty$. I've ...
0
votes
1answer
32 views

Hilbert's inequality for $\left|\sum_{n,m}a_n \bar a_m\right|$.

We know that, an Hilbert's inequality states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. Then write an inequality ...
0
votes
2answers
43 views

What is the general formula $t_n$ for this sequence? [closed]

$1, 4, 8, 13, 19, \dots$ What is the general formula or term for this sequence? How do you find it without a common ratio or difference? Please explain.
1
vote
1answer
47 views

Proving $(n+1)c^{1/(n+1)} - nc^{1/n} \le 1$ from first principles

Is it possible to prove that \begin{align*} (n+1)c^{1/(n+1)} - nc^{1/n} \le 1 \qquad c \in \mathbb{R}_+, n \in \mathbb{N} \end{align*} using only elementary techniques? (No calculus, no appeasement to ...
1
vote
1answer
28 views

Why is the identity function from $\Bbb R$ with the Euclidean metric to $\Bbb R$ with the discrete metric not continuous?

Using only the definition of sequential continuity, show an example that $f(x) = x: \Bbb R \to \Bbb R'$ is not continuous, where $\Bbb R'$ has the discrete topology. So the definition of ...
0
votes
1answer
26 views

Is it possible to exhibit a collection of sets

Let a subset $D$ of the natural numbers be called convergent or divergent when the associated series $\sum_{d \in D} \frac{1}{d}$ converges or diverges. Define a topology on $\Bbb{N}$ by defining the ...
2
votes
0answers
43 views

For which values of $\alpha, \beta, x \in \mathbb{R}, x \geq 0$, does the series $\sum_{n=1}^\infty n^ \alpha x^{n^{\beta}}$ converge?

I have to study, for $\alpha, \beta, x \in \mathbb{R}$, $x \geq 0$, the convergence/divergence/irregularity (i.e., when the limit of the $N$-th partial sum does not exist) of the following series: ...
1
vote
2answers
71 views

For which $x \in \mathbb{R}$ does the series $\sum_{n=1}^\infty \frac{x^n}{n!}$ converge?

One problem of my exercise book asks for which $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \frac{x^n}{n!}.$$ The answer given by the exercise book is $|x|\leq 1$, but ...
6
votes
1answer
52 views

For which $x\in \mathbb{R}$ does $\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$ converge?

I have to study for which values of $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$$ I was only able to say that the ...
5
votes
1answer
77 views

For what values of $x$ does the series $\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$ converge?

I have to study the values of $x$ for which $$\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$$ converges. First we say that we must have $x>0$. Then, I have started by rewriting the series as ...
1
vote
4answers
70 views

Proving that if $f$ is continuous then $f(a_n)$ converges

Suppose $f$ is continuous on the closed interval $[0,1]$ such that $f(0)=f(1)$. Prove that if $$(a_n)=\left(\frac{n(1+\cos(\pi n))}{2 n+1}\right)$$ then the sequence $(f(a_n))$ converges. I noticed ...
1
vote
2answers
42 views

Limited partial sum of $\displaystyle \sum _{n=1} ^{k} \cos(nx)$ are limited?

I'm wondering if it's true that $\displaystyle \sum _{n=1} ^{k} \cos(nx)$ has limited partial sum. I know it has representation $\displaystyle ...
6
votes
4answers
49 views

For which values of $\alpha \in \mathbb{R}$, does the series $\sum_{n=1}^\infty n^\alpha(\sqrt{n+1} - 2 \sqrt{n} + \sqrt{n-1})$ converge?

How do I study for which values of $\alpha \in \mathbb{R}$ the following series converges? (I have some troubles because of the form [$\infty - \infty$] that arises when taking the limit.) ...
0
votes
2answers
60 views

prove or disprove convergence [duplicate]

Im trying to prove or disprove the following , but I am having a hard time. It seems that the statement is true , but I have no idea how to prove it. If \begin{equation*} \sum_{n} a_{n}^2 ...
2
votes
1answer
39 views

How to tell if a series diverges or is indeterminate? Study of some cases of $\sum_{n=1}^\infty3^n (1+\frac{1}{n})^{n^2}k^n$

Suppose we have a series dependent on a parameter. For example: $$\sum_{n=1}^\infty3^n (1+\frac{1}{n})^{n^2}k^n.$$ By root test, we know that this series absolutely converges (hence converges) if ...
0
votes
2answers
58 views

Uniformly cauchy sequences

A sequence of functions $f_n$ is said to be uniformly cauchy if $$\forall \varepsilon > 0 \ \exists N > 0 :\forall z , \forall r, s > N: |f_r(z) - f_s(z)| < \varepsilon$$ How can I show ...
-4
votes
1answer
56 views

Is this series computable? [duplicate]

I would like to compute the value of this series: \begin{equation*} \sum_{n = 0}^{+ \infty} n . e^{- \alpha n} \end{equation*} Where $\alpha$ is a constant.
1
vote
0answers
34 views

Arithmetic progression Find first term and common difference when sum of 10 terms and the 8th term is given

Sigma is a car company that sell cars. Sigma sells $x$ cars in the first month and its sales increase constantly by $y$ cars every subsequent month. It sells $96$ cars in the $8^{th}$ month and the ...
-3
votes
1answer
32 views

Define $(a_n)_{n\in \mathbb{N}}$ by $a_n:=\prod_{k=2}^n (1-\frac{1}{k^{1+c}})$. Does $\lim_{n\to\infty} a_n=0$ hold? [closed]

Let $c\in \mathbb{R},c>0$ and define the sequence $(a_n)_{n\in \mathbb{N}}$ by $a_n:=\prod_{k=2}^n(1-\frac{1}{k^{1+c}})$. Does $\lim_{n\to\infty} a_n=0$ hold?
0
votes
2answers
46 views

Transformation of a function into a power series [closed]

How can I transform the real functions $\frac{1}{1-\sin x}$ and $\frac{x}{e^x-1}$ into power series with $x_0=0$?
2
votes
1answer
39 views

Series of gamma function with fixed real part and increasing imaginary part

I'm trying to evaluate the series or to pursue a upper bound, theoretically or numerically: $$ \sum_{k \ge 1} \left| \Gamma(m+2\pi ik/\log q) \right| $$ I know this series is convergent because each ...
2
votes
2answers
54 views

Sum of the series $\tan^{-1}\frac{4}{4n^2+3}$

Find the value of $$\sum^{n=k}_{n=1}\tan^{-1}\frac{4}{4n^2+3}$$ I tried multiplying numerator and denominator by $n^2$, but got nothing. How do I split the term inside $\tan^{-1}$?