Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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15 views

Is the sum of partial sums $\frac{2\pi}{k}-{k \mod 2\pi}$ bounded?

Suppose, we are given sequence $a_k= k\mod 2\pi$ and sequence $y_{k,n}$, where $y_{k,n},0\leq n\leq k$ is just sequence $x_0,\dots,x_k$ sorted in increasing order for a given integer $k$. What can we ...
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1answer
12 views

weakly convergent and the inner product in $l^{2}$

A sequence $\mathbf{x}^{(n)}$ in $l^{2}(\mathbb{R},\mathbb{N})$ converges weakly to $\mathbf{x}$ if, and only if, the sequence of inner products $(\mathbf{x}^{(n)} - \mathbf{x}, a)$ converges to $0$ ...
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1answer
50 views

Show that $a_n:=\frac{(-1)^{n-1}}{2n-1}$ converges

Show that $a_n:=\frac{(-1)^{n-1}}{2n-1}$ converges. If $(a_n)$ converges, the sequences is a Cauchy sequence. Which means: $\forall \epsilon >0 \,\,\,\exists N \in \mathbb{N}\,\,\,m,n>N:\left|\...
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2answers
79 views

Does the series $ \sum_{n=1}^{\infty}(x/e^x)^n$ converge?

Does the series converge for $x\in [0,\infty)$? I used the ratio test & get out that the series converges for whatever $x$ was.
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4answers
55 views

Reference for “If $(a_n)$ is a sequence of real numbers with $\lim a_n^2=0$, then also $\lim a_n=0$”?

Could you please give me a good reference as to the below-quoted real analysis result? If $(a_n)$ is a sequence of real numbers with $\lim a_n^2=0$, then also $\lim a_n=0$
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0answers
50 views

Show $\sum_{k=1}^n {k+1\choose 2}{2n+1\choose n+k+1}={n\choose 1}4^{n-1}$

I've been attempting to show that: $$\sum_{k=1}^n {k+1\choose 2}{2n+1\choose n+k+1}={n\choose 1}4^{n-1}\\ \sum_{k=2}^n {k+2\choose 4}{2n+1\choose n+k+1}={n\choose 2}4^{n-2}$$ Can anyone give some ...
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0answers
38 views

how to check if a number is of the form $a^b - 1$ [closed]

how to check if a number is of the form $a^b - 1$? For example 7 is of the form $$a^b - 1,$$since of the form $$2^3 - 1=7$$
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1answer
16 views

Convergence of summable Fourier series.

Let $f$ be a function on $[0,2\pi)$ satisfying $(\hat{f}(k))_{k\in \mathbb{Z}}\in l^1(\mathbb{Z}),$ where $$\hat{f}(k)=\int_0^{2\pi}f(t)e^{-ikt}dt.$$ Is it always true that $$f(x)=\sum_\mathbb{Z}\...
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30 views

If $-\sum a_n<\infty$, what could be $\sum a_n? $ $a_n>0.$ [closed]

I am wondering what could be $$\sum a_n, $$ if $$-\sum a_n<\infty.$$ The sequence $a_n$ is nonnegative. Is there any real value of $\sum a_n? $,
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109 views

A problem posed by Ramanujan involving $\sum e^{-5\pi n^2}$

While going through the list of problems posed by Ramanujan in Journal of Indian Mathematical Society I came across this problem involving theta functions: Prove that $$\frac{1}{2}+\sum_{n=1}^{\...
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2answers
38 views

Closed form of $\sum_{n=1}^{\infty}a^n \cos(nt)$

I'm trying to find a closed form (if one exists) for $$\sum_{n=1}^{\infty}a^n \cos(nt)$$ where $a \in (0, 1)$. I know that $$\begin{matrix} \sum_{n=1}^{\infty} \frac{a^n}{n} \cos(nt) &= &\...
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1answer
19 views

Find the sequence $a_n$ so that $\sum_{n=1}^{\infty} a_nsin(nx) = f(x)$ where $f(x)$ is a piecewise function.

Trying to solve a problem I reached a point where I know that $$\sum_{n=1}^{\infty} a_nsin(nx) = f(x) \text{, where }f(x) = \begin{cases} x & 0 \leq x \leq \frac\pi2 \\[5pt] \pi - x & \frac\...
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2answers
45 views

How do I find the sum of a power series $\underset{n=3}{\overset{\infty}{\sum}}\frac{x^n}{(n+1)!n\,3^{n-2}}$?

I have found the area of convergence to be $ x \in (-\infty, \infty)$, and this is how far I had gotten before getting stuck: $$ \begin{aligned} \sum_{n=3}^{\infty} \frac{x^{n}}{(n+1) ! n 3^{n-2}} &...
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1answer
16 views

Convergence of sequence uniformly in $n$ [closed]

Let $\{x_{n,m}\}$ be a sequence depends on two variables $(n, m)$. If we want to show that $\{x_{n,m}\}$ converges to $0$ when $m$ tends to $+\infty$ uniformly in $n$. What should show?
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40 views

What is the sum of the series??? [duplicate]

What is the sum of this series: $$S(z) = \sum_{n=1}^{\infty}{n^2z^n}$$ I know it diverges to infinity. Does the sum just equal infinity then?
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2answers
26 views

A Map is continuous on the inverse image of the set $(-\infty,r]$. Does this inverse image a closed set?

Let $U$ be a topological space and a map $g:U\to \mathbb{R}$. For a given $r\in\mathbb{R}$, define $E:= \{x\in U: g(x)\leq r\}$. If $g$ is continuous at every point of $E$, then Is it true that $E$ is ...
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1answer
22 views

Fourier Coefficients of Eisenstein Series $G_{2k}(\tau)=\sum\limits_{(m,n)\in\Bbb Z^2\setminus \{(0,0)\}}\frac{1}{(m+n\tau)^{2k}}$.

Suppose $\tau\in\Bbb C$ and $\Im(\tau)>0$. Also, let $k\in\Bbb Z_{>2}$, and $A=\Bbb Z^2\setminus\{(0,0)\}$. Then the Eisenstein series $G_{2k}(\tau)$ is given by $$G_{2k}(\tau)=\sum_{(m,n)\in A}\...
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1answer
51 views

An Interesting Property Concerning a Sequence of Integers

A non-decreasing sequence of positive integers $a_1,a_2,\dots a_n\ (n\geq 3)$ is good if for each $3\leq k\leq n$ there are $1\leq i\leq j<k$ such that $a_i+a_j=a_k$. Let $\ell,m$ be positive ...
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6 views

why if $h_{k}=f^{'}_{n_p}$ then $2\leq k\leq p$

I read article An elementary proof of Komlós-Révész theorem in Hilbert spaces by Guessous Mohamed (see page 328). But I don't understand why : if $h_{m+j}=f^{'}_{n_p}$ then $2\leq k\leq p$
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2answers
25 views

When is a series of matrices divergent. How to define divergence in this case?

In Quantum Mechanics we deal with series of operators represented as matrices like $$e^A = 1+ A + \frac{A^2}{2} + \dots$$ and similarly for $\sin(A) $, etc., where $A$ is a matrix. Now my question ...
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28 views

Calculating general term and sum of series

Recently while performing some calculations I came across this series $0\cdot1+1\cdot2+2\cdot4+3\cdot8+4\cdot12+5\cdot16+\cdots+9\cdot1$ Instead of calculating the sum, I thought of sum trick so I ...
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0answers
36 views

How to solve this infinite summation $\sum\limits_{r=0}^\infty\frac{2^r}{5^{2^r}+1}$? [closed]

If $\displaystyle \ell=\lim\limits_{n\to\infty} \sum\limits_{r=0}^n\frac{2^r}{5^{2^r}+1}$, then $24\ell$ is I tried it with writing the value of each term but there is not any pattern . I am unable ...
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1answer
57 views

How to find the $n$-th number whose sum of digits is $9$? [closed]

How to find the $n$th number whose sum of digits is $9$. First few numbers whose sum of digits is $9$ are $$9\quad 18 \quad 27 \quad36 \quad45 \quad54 \quad63 \quad72 \quad81\quad 90 \quad 108\quad ...
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1answer
69 views

Uniform convergence of $\sum _{n=1}^{\infty} \frac{\left(-1\right)^{n-1}}{n}x^n $

I treid to show that $$\sum _{n=1}^{\infty} \frac{\left(-1\right)^{n-1}}{n}x^n = \log(1+x)$$ for $\mid x\mid <1$ by showing that $$\sum _{n=1}^{\infty} (-1)^{n-1} x^{n-1} = \frac {1}{x+1}$$ ...
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1answer
33 views

Finding the poles of a certain Zeta function

I'm interested in the following zeta function: $$\zeta_D(s)=\sum_{(m,n)\in \mathbb N^2} \frac{1}{(m^2+n^2)^s}$$ Where naturally $(m,n)\neq (0,0)$. I'm mainly interested in its poles (so I can ...
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1answer
21 views

General form of the open mapping theorem

Let $X,X_1,X_2,...$ be real valued random variables on the same probability space $(\Omega, \mathcal{F},\mathbb{P})$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function.We know that ...
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1answer
24 views

Moment generating function from probability mass function

We are given the pmf: $$f_X(k) = \frac{1}{k(k+1)}, k \geq 1 $$ and we have to compute the moment generating function. So far I've got: $$M_X(t) = E[e^{tx}] = \sum_{k=1}^{\infty} e^{tk} \frac{1}{k(k+1)}...
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0answers
14 views

How do I prove or disprove that the series are Fourier transformation for some function: $\sum^{\infty}_{k=1} \frac{\cos 2k x}{\sqrt{2k \ln 2k}}$

Let's say, there is a sum $\Sigma^{\infty}_{k=1} \frac{\cos 2k x}{\sqrt{2k \ln 2k}}$, how do I prove or disprove that it is a Fourier transformation for some function? Any ideas or online sources are ...
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1answer
26 views

Polynomial-Exponential Inequality

Consider the following sequences $$ x_n = \sup \{ k \in \mathbb{N} : e^{2^k} 2^k \leq n \} $$ $$ y_n = \sup \{ k \in \mathbb{N} : e^{2^k} 2^{2k} \leq n \} $$ Clearly $x_n \geq y_n$ but I ...
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2answers
58 views

Find an equivalent sequence as $n\to +\infty$ of $u_1>0, u_{n+1} = \frac{u_n}{n} + \frac{1}{n^2}$

Let $u_1>0$ be a real number. Let us consider $(u_n)_{n\geq 1}$ the sequence such as: $$ \forall n \geq 1, u_{n+1} = \frac{u_n}{n} + \frac{1}{n^2}\quad (\star) $$ Find an equivalent of $u_n$ as $...
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0answers
6 views

Leading order solution for integral of $e^{z\sin(\omega t)}$

I am trying to show that $\int e^{z\sin(\omega t)}dt = I_0(z)+O(\frac{1}{\omega})$ using a Bessel function expansion. Using the expansion $e^{z \sin(\omega t)}=I_0(z)-2\sum_{k=1}^\infty(-1)^k\bigg(...
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2answers
56 views

Evaluating: $\lim_{n\to \infty} \sum_{i=1}^n \frac{1}{\frac{an}{b-a}+i}$

$$\lim_{n\to \infty} \sum_{i=1}^n \frac{1}{\frac{an}{b-a}+i},\ a,b\in\Bbb R\setminus\{0\},\ a\ne b$$ I know for a fact that the solution can be found via Laurent Series if that hint helps. I inserted ...
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1answer
40 views

I have to find the marginal pmf's $f_X$, $f_Y$ and $f_{X+Y}$ of $X+Y$.

I have to find $D > 0$ such that $f(x, y)$ = $D$($\frac{1}{x+y−1}$$+$ $\frac{1}{x+y+1}$ $−$ $\frac{2}{x+y}$) is the joint pmf $f_{X,Y} (x, y)$ of a random vector $(X,Y)$ in {$1,2,..$}$^2$. And ...
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3answers
31 views

sum upto n terms where rth term is $r(r+1)2^r$

sum upto n terms where rth term is $r(r+1)2^r$. I tried to make a telescoping series but failed.It seems like i have to subtract and add something from r(r+1) such that power of 2 also change.Is ...
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0answers
8 views

Assigning Probabilities to Alarming Events for predicting System Outages

I have a problem. I have two sets of data, let's assume they are clean and well-organised. Data-set 1: Alarm Events. Data-set 2: System Outages. The two data-sets have only one feature in common - the ...
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0answers
20 views

How to prove or disprove that the series are Fourier transformation for some function.

Let's say, there is a function $\Sigma^{\infty}_{k=1} \frac{\cos 2k x}{\sqrt{2k \ln 2k}}$, how do I prove or disprove that it is a Fourier transformation for some function?
2
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2answers
46 views

power series $\large{\Sigma_{n=0}^{\infty}} \frac{(n!)^2 x^n}{(2n)!}$, Radius of convergence

I have the power series $$\large{\Sigma_{n=0}^{\infty}} \frac{(n!)^2 x^n}{(2n)!}.$$ I found the radius of convergence to be $(-4,4)$ using d'Alembert rule. Now I am trying to find what is happening ...
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0answers
13 views

I need to find the convergence interval for a trigonometric series.

$$\Sigma^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \cot^n x} \le \frac{1}{n(n+1) \cot^n x} = \left[ y = \cot^n x \right] \\ \text{Due to necessary condition for convergence: } ...
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0answers
66 views

Evaluate: $\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$

The questions asks to evaluate: $$\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$$ The answer provided is $-\frac{1}{4}$, but I don't know how to solve it. I am not sure how to ...
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3answers
28 views

Tried to apply the ratio test to determine the convergence interval, but get the limit as a constant.

So here is the series: $\Sigma_{n=1}^{\infty} \frac{x^{2n}}{1+x^{4n}}$ $$\left| \frac{x^{2(n+1)} (1+x^{4n})}{x^{2n}(1+x^{4(n+1)})}\right| = \left|\frac{x(1+x^{4n})}{1+x^{4n+4}} \right| \overset{\text{...
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1answer
25 views

Given 2 numbers A and B how to check if (A+B) > (A*b)? [closed]

Given $2$ numbers $ A $ and $B$ how to check if $(A+B) > (A\cdot B)$, where $+$ is concatenation ? e.g. $A = 10, B = 2$. So $ A + B = 102$ and $A \cdot B = 20$ so Here $(A+B) > (A\cdot B)$
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1answer
35 views

Is there a formula for finding the sum of square roots of first n natural numbers? [duplicate]

$$\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{5} ....... N,terms $$ Is there a formula for finding the sum of square roots of first n natural numbers?
2
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3answers
90 views

Find $\displaystyle\lim_{n\to \infty} \int_0^1 nx^n e^{x^2} dx$

The value of $\displaystyle\lim_{n\to \infty} \int_0^1 nx^n e^{x^2} dx$ is ____________(round off to three decimal places) I tried integrating by parts and bring out some recurrence relation , but ...
2
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2answers
52 views

Evaluate $\sum_{n=1}^{50}n\cdot n!$. [duplicate]

Evaluate $\displaystyle\sum_{n=1}^{50}n \!\cdot\! n!$ I tried to write the sum like: $1+2\!\cdot\!2!+3\!\cdot\!3!+4\!\cdot\!4!+5\!\cdot\!5!+\ldots=1+4+18+96+600\ldots$ I can find a recursion like $...
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1answer
31 views

Consider the sequence $u_n=\sum_{r=1}^{n}\frac{r}{2^r},n≥1$. Find the limit of $u_n$ as $n\rightarrow\infty$ [duplicate]

QUESTION: Consider the sequence $$u_n=\sum_{r=1}^{n}\frac{r}{2^r},n≥1$$ Find the limit of $u_n$ as $n\rightarrow\infty$ MY APPROACH: I thought that this question has been asked before. But I am a bit ...
2
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1answer
37 views

Polynomial-Exponential Equation

Consider the sequence $$ x_n = \sup \{ k \in \mathbb{N} : e^{ 2^{k}} 2^k \leq n \} $$ I'm wondering if it possible to deduce that there exists $\alpha \in (0,1)$ for which $ e^{2^{x_n}} = O(n^...
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3answers
58 views

What is the sum of n terms 1.5.9+5.9.13+9.13.17…?

How to solve this? My attempt is to write the $r^{th}$ term which is $$(4r-3)(4r+1)(4r+5)$$ Then let $p=4r+1$ The $r^{th}$ term becomes $$(p-4)(p)(p+4)=p^3-16p$$ Now we know summation $\sum p^3 =[\...
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0answers
16 views

Density of tensor products of eventually zero sequences.

Let $c_{00}(\mathbb{Z})$ denotes the space of eventually zero sequences over $\mathbb{Z}$. Let us define $$\mathcal{P}c_{00}(\mathbb{Z}^2)=span \{(a_k)_{k\in \mathbb{Z}^2 } : a_{(k_1,k_2)}=b_{k_1}c_{...
1
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3answers
34 views

How can i prove the convergence of this series [closed]

Supposing that $a_n$ is any sequence Knowing that$\displaystyle{\lim_{n \to \infty}n^4a_n=0}$, how can I show that $\sum_{n=1}^{\infty} a_n$ is convergent. Sorry but I really dont have any work to ...
1
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1answer
48 views

Comparing series $\ln(2)+\ln(\ln(2)+1)+\ln(\ln(\ln(2)+1)+1)+…$ to $\sum 1/k^2$

This is related to my previous question Does $\ln(1)+\ln(\ln(2)+1)+\ln(\ln(\ln(3)+1)+1)+...$ converge?. In the answers it is shown that $$\ln(2)+\ln(\ln(2)+1)+\ln(\ln(\ln(2)+1)+1)+...$$ diverges. In ...

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