Tagged Questions
4
votes
0answers
56 views
Limit of infinite sum
What is the answer to the following limit?
$$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$
4
votes
1answer
59 views
Calculate the limit of two interrelated sequences?
I'm given two sequences:
$$a_{n+1}=\frac{1+a_n+a_nb_n}{b_n},b_{n+1}=\frac{1+b_n+a_nb_n}{a_n}$$
as well as an initial condition $a_1=1$, $b_1=2$, and am told to find: $\displaystyle ...
0
votes
1answer
17 views
Approximating sums of powers of integers: why does $\sum_{i=1}^n i^r \div \frac{n^{r+1}}{r+1} \to 1$ as $n \to \infty$?
I know there are exact formulas for sums of integer powers of integers ($\sum_{i=1}^n i^r$ with $r \in \mathbb{N}$), but I was interested in approximating them. One way occured to me through a ...
1
vote
1answer
42 views
REVISTED$^1$ - Order: Modular Arithmetic
Relevant Literature:
Question:
Observe that $2^{10}=1024≡−1 \pmod{25}$.Find the order of $2$ modulo $25$.
Thoughts:
Direct answers are OK, but I'd like to know if I'm right that what I'm really ...
1
vote
2answers
51 views
Convergence of these series
$$\sum_{n=1}^\infty \frac{2nx^{n}}{(n+1)^{2}3^{n}} \tag{1}$$
$$\sum_{n=1}^\infty x^{n}\tan\frac{x}{4^{n}}\tag{2}$$
Is there any good article that describes an equivalents like if $$ ...
0
votes
1answer
57 views
Compute limit of the sequence given by $x_1=1$, $x_{n+1}^2=\frac{x_n+3}{2}$
Let $\left(x_n\right)$ be a real sequence such that $x_1=1,x_{n+1}^2=\dfrac{x_n+3}{2},\forall n\geq 1$.Compute $\lim_{n\to+\infty} 3^n\sqrt{\dfrac{9}{4}-x_n^2}$
3
votes
7answers
126 views
How to prove that $1/n!$ is less than $1/n^2$?
I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.
How to prove ...
0
votes
3answers
36 views
Sequence of the ratio of two successive terms of a sequence
If $(a_n)_{n\in N}$ is a strictly decreasing sequence of real number converging to $0$ and s.t. $\forall n\in N$, $0<a_n<1$, does the following limit:
$$
\lim_{n}\frac{a_{n+1}}{a_n}
$$
exists?
...
1
vote
3answers
54 views
Limit as N goes to Infinity
Consider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$
I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, ...
3
votes
1answer
39 views
Show that sequence approaches fixed point of a function
Problem
Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
3
votes
1answer
49 views
How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$?
How can we compute the following limit:
$$\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$$
Mathematica gives the answer $\sqrt{e}$. However, I do not know to do it.
2
votes
1answer
62 views
Solving the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$
Suppose the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$ as $f(t)=\frac{e^t}{t}$ and $n\in \mathbb{N} \setminus{0}$.
How to prove that:
The equation above has a unique solution $U_n$ ...
3
votes
3answers
51 views
Determine if a sequence converges using the number e
Knowing that the number $e =\lim_{n\to\infty}\left(1+{1\over{n}}\right)^n$ solve $a_n=\left({n+1}\over{n+3}\right)^n$
So (...)
$$\lim_{n\to\infty}\left({n+1}\over{n+3}\right)^n = ...
3
votes
1answer
54 views
To calculate $ \lim_{n\to \infty} \Big(\sum_{m=1}^r (a_m)^n\Big)^{1/n}$
Let $a_1 , a_2 , ..., a_r$ be positive real numbers such that $a_1 > a_2 > ... > a_r$. Without any more information given , can we exactly calculate $$ \lim_{n\to \infty} \Bigg(\sum_{m=1}^r ...
1
vote
1answer
39 views
Sequence - Convergence?
I have to proof the following:
$ \lim\limits_ {n\to\infty} \dfrac{\sqrt[n]{n!}}{n} = \frac{1}{e}$
Do you have any hints for me, since I do not know where to start..
1
vote
1answer
32 views
How to determine the Sum of a function series
The Task is to find the Sum of the given function series, that is defined as: $\sum\limits_{n=0}^{\infty}\frac{x^{4n+1}}{4n+1}$
I'm kinda lost, but at least i managed to take a few steps towards the ...
11
votes
6answers
280 views
Why is the definition of “limit” difficult to understand at first?
Tomorrow I teach my students about limits of sequences. I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier. But first I need to ...
3
votes
1answer
53 views
Limits with sums and integrals
It's one of my homework exercises that is rather problematic to me. Apparently the last thing to do is to squeeze it but I don't see yet how to do that. Could you help?
...
2
votes
2answers
87 views
Solve limits in Lebesgue integral
Solve the limits of below:
(1) $\lim\limits_{n \to \infty} \int_0^n (1+\frac{x}{n})^n e^{-2x}dx$.
(2) $\lim\limits_{n \to \infty} \int_0^n (1-\frac{x}{n})^n e^{\frac{x}{2}}dx$.
(3) ...
3
votes
1answer
39 views
Integral, limit, sequence of functions
I'm not sure how to formulate the title, but here is a problem I've come across recently and I'm not sure how to go about solving it.
Let $$P_n(x) = \frac{x^n(bx-a)^n}{n!}, \quad a,b,n \in ...
7
votes
3answers
97 views
Find the limit of $ x_n = \prod_{j=2}^{n} \left(1 - \frac{2}{j(j+1)}\right)^2$
I am stuck on the following problem:
Let $x_n=(1-\frac{1}{3})^2(1-\frac{1}{6})^2(1-\frac{1}{10})^2 \ldots...(1-\frac{1}{n(n+1)/2})^2, \text{where} \space n \geq 2$. Then $\lim_{n \to ...
2
votes
1answer
69 views
Computation of standard series
I am stuck on the computation of the following sum:
$\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?
6
votes
1answer
77 views
integration as limit of a sum
If $f$ is continuous on $[0, 1]$ then
$$\lim_ {n\to\infty}\sum_{j=0}^{\lfloor n/2\rfloor} \frac1{n}f\left(\frac {j}{n}\right) = ? $$
will the answer be that the limit exists and is equal to $ ...
16
votes
5answers
373 views
limit of : $a_{n+2} =\frac{1}{a_n} + \frac{1}{a_{n+1}}$
$a_n$ is a real sequence, $a_1,a_2$ are positive and for all $n>2$ : $$ a_{n+2} =\frac{1}{a_{n+1}} + \frac{1}{a_{n}}.$$
Prove that: $\displaystyle \lim_{n\to \infty} a_n$ exists, then find it.
...
2
votes
3answers
70 views
Show that $(x_n-y_n)$ converges to $x-y$.
Given $(x_n)$ and $(y_n)$ are sequences of real number which converge to $x$ and $y$ respectively. Show that $(x_n-y_n)$ converges to $x-y$.
If it's asking about $(x_n+y_n)$. I know that I can ...
1
vote
1answer
50 views
How find this intergral $\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy$
find the value
$$\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy,\qquad k\in N^{+}$$
my idea:
\begin{align}
...
3
votes
4answers
100 views
How to prove that $\displaystyle \lim_{n \to \infty} \sqrt[n]{n}=1$? [duplicate]
I have seen this fact thrown around a lot and never really stopped to prove it; plugging in a few values convinced me of its truth. But I would like to see the result proved. To be clear, this is not ...
0
votes
1answer
62 views
Lim sequence $\neq$ lim subsequence
Let $\{x_n\}$ be a sequence and $\{y_k\}$ be a subsequence of $\{x_n\}$. Can you show me an example of $\{x_n\}$ such that $\displaystyle \lim_{k \to +\infty} y_k= +\infty$, but $\displaystyle \lim_{n ...
1
vote
3answers
67 views
Finding the limit $\lim_{n\to \infty}\int^n_0 e^{-\lambda x}\mathrm dx$
Find the following limit:
$$\lim_{n\to \infty}\int^n_0 e^{-\lambda x}\mathrm dx$$
for all $\lambda>0$.
1
vote
1answer
44 views
Prove that for every $p>0$, $\lim_{n\rightarrow∞}\int_n^{n+p}{\sin x\over x} = 0$
Got stuck with this question:
Prove that for every $p>0$, $\displaystyle \lim \limits_{n\rightarrow∞}\int_n^{n+p}{\sin (x)\over x} = 0$.
Thanks in advance for any help!
0
votes
2answers
31 views
Investigate monotony, bound and convergence
I'm doing an exercise in the college, and I ran into a doubt: a friend says me that I've made a mistake, but I can't find it.
The exercise asks: "Investigate if the sequence $2^n\over{(n+1)!}$ is ...
1
vote
3answers
68 views
Show that the sequence $(x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.
Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.
I'm not familiar with proving divergent sequence. Do anyone have any des? ...
2
votes
2answers
64 views
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
1
vote
2answers
76 views
The mathematical and technical approach to a limit of a sum of a sequence
In Calculus, what is the most preferred mathematical and technical way to approach a limit of a sum of a sequence?
Take for example:
$$ \lim_{n \to \infty} {\frac{1}{n} \sum_{k=1}^{n} {\ln(1 + ...
1
vote
2answers
49 views
Prove or disprove a result for a double sequence.
Suppose that a double sequence $\{a_{n,k}\}=\left\{\frac{1}{n^{\frac{k-1}{k}}}\right\}$. Prove or disprove $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}a_{n,k}=0$.
1
vote
1answer
105 views
Infimum and limit
I was having trouble with the following question. Any help would be highly appreciated.
Let $A$ be the set of K-dimensional vectors with non-negative components. Let $B$ be the set of K-dimensional ...
2
votes
2answers
88 views
What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?
How to solve the following limit question?
$$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$
Thanks a lot.
3
votes
2answers
42 views
If the iterated limits are equal does the double sequence converge?
If $(a_{m,n})$ is a double sequence (in $\mathbb{R}$ or $\mathbb{C}$) and $\lim_m\lim_n a_{m,n}=\lim_n\lim_m a_{m,n}=a$ then can we deduce $\lim_{m,n}a_{m,n}=a$? More specifically can we deduce ...
3
votes
3answers
98 views
Finding a limit where binomial coefficients appear as powers.
This one is for my mate that is 2 years older than me. Could you help please?
$$\lim_{n\to\infty}\frac{\sqrt[\uproot{3}\Large 2^n]{n^{\textstyle\binom{n}{0}}\cdot ...
4
votes
2answers
97 views
Finding the infinite product $\prod_{n=1}^\infty\; \left(1+ \frac{1}{\pi ^2n^2}\right) $
How do I find:
$$\prod_{n=1}^\infty\; \left(1+ \frac{1}{\pi ^2n^2}\right) \quad$$
I am pretty sure that the infinite product converges, but if it doesn't please let me know if I have made an error.
...
3
votes
1answer
37 views
Limit of Multivariable Fourier Series
If I have some Fourier Series representation of a function with $x$ period of $2L$
$$G(x,y) = \sum_{n = 1}^{\infty} \left[a_n \sin\left(\frac{n \pi x}{L}\right) + b_n \cos\left(\frac{n \pi ...
5
votes
2answers
65 views
Which is the function that this sequence of functions converges [duplicate]
Prove that $$ \left(\sqrt x, \sqrt{x + \sqrt x}, \sqrt{x + \sqrt {x + \sqrt x}}, \ldots\right)$$ in $[0,\infty)$ is convergent and I should find the limit function as well.
For give a idea, I was ...
9
votes
3answers
140 views
Evaluating $\lim \limits_{n\to \infty} \left( n \int_{0}^{\frac \pi 2} 1-\sqrt [n]{\sin x} \,\mathrm dx \right)$
Evaluate the following limit:
$$\lim \limits_{n\to \infty} \;\; n \int_{0}^{\frac \pi 2} \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$
I have done the problem .
How I solved is
First I ...
4
votes
1answer
38 views
Find the common limit of $\frac{2}{1/a_n+1/b_n}$ and $\sqrt{a_n b_n}$
Let $a_0=1$ and $b_0=2$, then
\begin{align}
a_{n+1} &= \frac{2}{\frac{1}{a_n}+\frac{1}{b_n}}, \\
b_{n+1} &= \sqrt{a_n b_n}.
\end{align}
The sequences $(a_n)$ and $(b_n)$ converge to the same ...
1
vote
0answers
66 views
To show limit of $\left|\frac{ a_{n+1}}{a_n}\right|$ is smaller than lim inf $\left|(a_n)^{1/n}\right|$
Suppose that $a_{n}$ is a sequence of real numbers such that $\lim\limits_{n\to\infty} {\frac{\left|a_{n+1}\right|}{\left|(a_n)\right|}}$ exists,
then $$\lim\limits_{n\to\infty} ...
3
votes
2answers
89 views
Find the limit of the sequence $\frac{n!}{2^n}$ as n tends to infinity
We've only been taught to find limits using the Squeeze Theorem and L'Hopitals Rule, so I'm not sure how to go about finding the limit of this sequence.
1
vote
1answer
63 views
Find the supremum of $\left ( n+1 \right )^{\frac{2}{n^2}}$
As in the topic, my task is to find supremum and infimum of a given set $$f(n):=\left ( n+1 \right )^{\frac{2}{n^2}}, n\in \mathbb{N}$$What is funny, I managed to do this task few weeks ago and I ...
1
vote
2answers
47 views
Convergent sequence question…
I have a homework question that I am not sure how to begin. We are asked to suppose $\{a_n\}$ and $\{b_n\}$ are sequences such that $\{a_n^2 + b_n^2\} \rightarrow 0$.
We have to prove $\{a_n\} ...
1
vote
2answers
65 views
How to show that $f$ is continuous only at $x=0$
Can any one help me to answer this question:
Assuming $$f(x)=\begin{cases} x &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$
show that $f$ ...
1
vote
2answers
38 views
If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.
A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$
Let $(X,d)$ be a metric space and let ...
