2
votes
1answer
57 views

Calculate $\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$

Let $\alpha$ be a positive number. Calculate $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$ Edit: I have deleted my attempt, it didn't seem to lead me anywhere and I discovered ...
3
votes
2answers
39 views

Differentiability for the uniform limit of a uniformly bounded sequence of functions

Let a sequence $\{f_n\}\subset C^1(\mathbb{R})$ and $f\in C(\mathbb R)$ such that $f_n \to f$ uniformly and $f_n, f'_n$ are uniformly bounded. Question : is $f \in C^1(\mathbb R)$ ?
3
votes
2answers
53 views

What is the value of the limit $\lim_{a\searrow 0}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{a}}$?

Clearly the series $$ \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{a}} $$ converges (conditionally), as an alternating series of as absolutely decreasing sequence, for all $a>0$. The question is: What ...
0
votes
2answers
26 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
1
vote
1answer
60 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
4
votes
1answer
41 views

Find $\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right)$

Find $$\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right).$$ I have tried rewriting the sum in a clever way, applying the Mean Value Theorem or Stolz-Cesaro ...
0
votes
1answer
31 views

Relation between limit of functions and sequences

I need to prove that the sequence $$ \frac{\sqrt{n}}{\log n}$$ diverges. I know that $$\lim \frac{\sqrt{x}}{\log x} = \infty$$. Is there any theorem that relations the limit of the function with the ...
1
vote
1answer
73 views

interval of convergence of $e^x$

Can somebody explain how to find the interval of convergence for $$ e^x=\sum_{n=1}^\infty\frac{x^n}{n!} $$ I don't fully understand the ratio test/root test/integral test etc. and I don't understand ...
1
vote
3answers
61 views

Using Riemann sums to show that $\sum_{i=1}^n 1/i = \log{n} +c+O(1/n)$

I want to show that there exists a constant $c$ such that: $$ \sum_{i=1}^n 1/i = \log{n} +c+O(1/n) $$ I am thinking about Riemann sums. Any hints on that?
3
votes
3answers
69 views

For what values of $p$ is the series $\sum_n \frac{1}{n\ln(n)^{p}}$ convergent?

The series is: $\displaystyle\sum_{n=1}^\infty \dfrac1{n (\ln n)^p}$ I don't know what to do from here since $p$ is on $\ln$. Would $p$ still have to be $> 1$ since $\ln$ is changing in terms of ...
3
votes
2answers
53 views

For what values of $x$ does the geometric series $(2+x)+(2+x)^2+(2+x)^3+\cdots$ converge?

I am stuck on this geometric series question: For what values of $x$ would the infinite geometric series $(2+x)+(2+x)^2+(2+x)^3+\cdots$ converge? Formula for series: $$S_n=\frac{a(1-r^n)}{1-r}$$ ...
2
votes
1answer
27 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
4
votes
2answers
93 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
0
votes
1answer
63 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
2
votes
3answers
61 views

Finding the second derivative of an infinite series

I'm asked to find the 2nd derivative of $$f(x)=-2x+\frac{2x^3}{3!}-\frac{2x^5}{5!}+\cdots+(-1)^{n+1}\frac{2x^{n+1}}{(2n+1)!}+\cdots=\sum \limits_{n=0}^{\infty} \frac{(-1)^{n+1}2x^{2n+1} }{(2n+1)!}$$ ...
1
vote
1answer
26 views

Recovering a continuous function from a discrete one.

Consider a well-behaved function $f(x)$ defined on $x\geq0$, and construct a discretized version of it using the Dirac-delta function: $$ ...
1
vote
3answers
57 views

How to calculate the integral of $\sum_{n=1}^\infty (1/r)^{n+1} r^2$?

How to calculate this integral? $$\int\limits_0^1 {\sum\limits_{n = 1}^\infty {\left( {\frac{1}{r}} \right)} } ^{n + 1} r^2 dx$$ Here $r$ is a real number
1
vote
0answers
23 views

'Deriving' the Laplace Transform from the z Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
4
votes
5answers
173 views

Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent?

Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$ Attempt: I got this using Ratio Test: $$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
0
votes
3answers
46 views

Question on Sequences and limits

If sequence {$a_n$} satisfies $\displaystyle \lim_{n \to \infty} (2n-1)a_n=40$, what is the value of $\displaystyle \lim_{n \to \infty}na_n$ ? Any hints ?
6
votes
0answers
62 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
1
vote
2answers
102 views

Does This Function Exist?

I am trying to construct a piecewise function $g:[0,1] \rightarrow \mathbb{R}$ with $g(0)=0,\hspace{3mm}$ $g \geq 0$, $\hspace{3mm}$ $\int_0^x g(t)dt\leq x,\hspace{3mm}$ and such that there is a ...
1
vote
1answer
26 views

Build a sequence, $a_n$ with $PL=\{0,1,2\}$

I was asked to build a sequence which has exactly three partial limits: $\{0,1,2\}$. Also, for every $n\in\Bbb{N}: \left|a_{n+1} - a_n\right| < 1$ At first I thought about: $$a_n = \begin{cases} ...
1
vote
1answer
85 views

Explanation of the formulas for sums $\sum nr^n$ and $\sum n^2 r^n$

So I am studying series for an exam right now and there is an example in the book I am studying (unfortunately the book is specific to my university so I cannot give any link) where certain series' ...
1
vote
0answers
69 views

Why is sequence $(1+\frac{1}{n})^{n+1}$ descending? [duplicate]

I was studying the proof of $e$ number when I noticed something: Why is the sequence $(1+\frac{1}{n})^{n+1}$ descending? It starts ascending with grater n but in one moment it starts descending? Why ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
2answers
55 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) $\sum_{n=0}^{\infty}c_{n}2^n$ ...
4
votes
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) - ...
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
vote
1answer
64 views

An upper bound on a sequence of positive numbers $x_n$ such that $x_{n+1} \le \min \{b \cdot x_n,c\}$

Suppose $\{x_1, x_2,\ldots, x_n,\ldots \}$ is a sequence that satisfies $x_0 = a$, and $x_{n+1} \le \min \{b \cdot x_n,c\}$, where $a,b,c>0$ are constant given numbers, and $x_i>0$ for ...
7
votes
0answers
63 views

A cotangent series related to the zeta function

$$\sin x = x\prod_{n=1}^\infty \left[1-\frac{x^2}{n^2\pi^2}\right]$$ If you apply $\log$ to both sides and derivate: $$\cot x = \frac{1}{x} - \sum_{n=1}^\infty \left[\frac{2x}{n^2\pi^2} ...
7
votes
2answers
116 views

Limit of $\lim\limits_{n\to\infty} (1 + \frac{x_n}{n})^n$

Many websites and calculus books give this well known result \begin{equation} \lim\limits_{n\to\infty} \left(1 + \frac{x}{n}\right)^n = e^x \end{equation} However, a textbook I was reading casually ...
2
votes
1answer
56 views

Asymptotics of coefficients $[x^n] \frac{1}{\Gamma(1+x)}$ as $n$ is great

I am interested in the behaviour, as $n$ is great, of the coefficients $g_n$ in the Maclauren expansion of $\displaystyle \frac{1}{\Gamma(1+x)} $. We have $$ \frac{1}{\Gamma(1+x)}=\sum_{n=0}^\infty ...
3
votes
2answers
40 views

Squeeze Principle

Let $\{a_n\}$ be a sequence of positive integers and let $f$ be a function on the integers. Suppose that for each $\epsilon \in (0,1)$ there exists an integer $L$ such that for every $n \geq L$ we ...
4
votes
2answers
87 views

Convergence of tetration sequence.

This question arose from here. I am interested to find a nice proof about the convergence of $${^n}a=\underbrace{a^{a^{\ .^{\ .^{\ .^a}}}}}_{n\ \text{times}}.$$ I find with google a necessary and ...
9
votes
5answers
256 views

Prove that $\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx=\frac{16\sqrt{3}}{729}\pi^5+\frac{605}{54}\zeta(5)$

This integral comes from a well-known site (I am sorry, the site is classified due to regarding the OP.) $$\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx$$ I can calculate the integral using the help of ...
2
votes
2answers
67 views

Can we expect to find some constant $C$; so that, $\sum_{n\in \mathbb Z} \frac{1}{1+(n-y)^{2}} <C$ for all $y\in \mathbb R;$?

Fix $y\in \mathbb R;$ and consider the series: $$\sum_{n\in \mathbb Z}\frac{1}{1+(n-y)^{2}}.$$ My Question is: Can we expect to find some constant $C$; so that, $$\sum_{n\in \mathbb Z} ...
2
votes
1answer
71 views

inverse quadratic infinite sum [duplicate]

I have the following infinite sum $$\sum_{n=1}^{\infty} {\frac{1}{n^2+a^2}}=\frac{1}{2a^2}\left(1+a\pi\coth{a\pi}\right)$$ I would like to know how to prove it in a variety of ways, so any proofs ...
0
votes
0answers
38 views

Why there's no articles about the eta function convergence?

I've been searching about a proof that the eta function converges for $\mbox{Re}(z)>0$ but the ONLY page I've found that claims to prove it was in this question: ...
1
vote
1answer
41 views

$\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}\leq C$ for all $y\in \mathbb R$?

Fix $y\in \mathbb R$ and $s>1.$ Consider the series: $$I(y)=\sum_{n\in \mathbb Z} \frac{(1+n^{2})^{s}}{1+(n-y)^{r}}.$$ My Question is: Can we choose $r$ large enough so that $I(y)< C$ for ...
1
vote
2answers
37 views

Multplication of series

My textbook is taking about the Cauchy product and I don't quite understand it and it says that when multiplying series, the sum of the third one is equal to the product of the sums of first two ...
1
vote
2answers
83 views

Series $ \sum_{n = 0}^\infty \frac{(-1)^nsin(n)}{n!} $ is absolutely convergent?

I'm having trouble proving the series $$ \sum_{n = 0}^\infty \frac{(-1)^n\sin(n)}{n!} $$ is absolutely convergent. My try I know that the series $$ \sum_{n = 0}^\infty \frac{\sin(n)}{n!} $$ ...
3
votes
4answers
108 views

Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?

Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge? Since the function $x+1/x$ is strictly monotonic increasing for all $x>1$, I don't think that the limit converges, ...
0
votes
3answers
113 views

Computing the sum of an infinite series

I am confused as to how to evaluate the infinite series $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}$$. I tried splitting the fraction into two parts, i.e. ...
1
vote
1answer
48 views

Taylor series and Lagrange's remainder f(x)=$e^x$

In my textbook the Lagrange's remainder which is associated with the Taylor's formula is defined as: $R_{n}(x)= \frac{(x-a)^n}{n!} f^{(n)}(a + \vartheta (x-a))$, for some $\vartheta$ $\in$ <0 ,1> ...
3
votes
1answer
316 views

How to find the tenth derivative of an exponential function?

I have this equation, $f(x) = e^{-x^2}$. My question is how should I find $f^{(10)} (0)$, ie the tenth derivative of this equation. I have tried differentiating to get a formula, and I get ...
8
votes
0answers
94 views

A closed form for $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ ...
3
votes
3answers
130 views

Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.