Recurrence relations, convergence tests, identifying sequences
0
votes
2answers
39 views
Taylor series of $f(x)=\frac {e^x-1}{x}$
I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions.
How to simplify the function so that it can be expanded more easily?
1
vote
1answer
40 views
Math arithmetic
So I got this question in my exam yesterday, and only a few people from my school could solve this.
What's so hard about it? I have learned the simple arithmetic questions, but I don't really ...
2
votes
3answers
33 views
find the condition on A for the summation to be convergent
The summation is:
$$\sum_{n=1}^\infty \frac{ \sqrt { n + 1 } - \sqrt n }{n^A}$$
I don't know how to even begin. Hints??
0
votes
0answers
38 views
funcitonal series convergence… SOS… [duplicate]
Is it possible to show that $f(x) = \sum_{k=0}^\infty (-1)^k \frac{x^k \sqrt{k}}{k!}$ converges even when $x\rightarrow \infty$ ?
i.e. As a function of x, does $\lim_{x\to\infty} f(x)$ exist or at ...
-1
votes
1answer
33 views
Alternating functional Series Convergence SOS…
Does the following series converge?
$\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$
what is the radius of convergence?!!
1
vote
1answer
26 views
How do I write the generic series definition that can produce any term in the series expansion?
I'm trying to learn how to build the generic series definition for a series of numbers. For some reason I'm having a hard time pulling out this pattern. I'm always pulling out the wrong details for ...
0
votes
1answer
27 views
Extracting distinct sequence
Let $(x_n)$ be a non-constant sequence in $\mathbb{R}$ and $x_n\rightarrow p$ for some $p\in \mathbb{R}$. Can I always extract a subsequence $(x_{n_k})$ whose all elements are distinct? (of course ...
16
votes
1answer
159 views
Prove every integer exists in this sequence?
Please could someone give me a hint on this sequences question? The question is to prove that every integer appears infinitely many times in the following sequence:
$$ \pm 1^{2} , \pm 1^{2} \pm 2^{2} ...
1
vote
2answers
36 views
“Translating” one value $- \infty$ to $+ \infty$ to another ($+ \infty$ to $ \gt 0$)
Well even if I need to use the following in a computer game this is a math question. I have a world map which I can scroll with a scroll velocity $(f(x))$ with my mouse. And I have a zoom factor $(x)$ ...
0
votes
0answers
22 views
Calculation the variance of the forecast error?
Hi there stuck on the following:
Consider the model: $$y_{t}=(1+a)y_{t-1}-(a)y_{t-2}+\epsilon_{t}$$
where $\epsilon_{t}$ is a white noise problem:
1) Transform $y_t$ into some other series $w_t$ ...
1
vote
0answers
29 views
show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle [duplicate]
I need to show $\sum_{n=0}^{\infty}{z^n\over n}$ is convergent on the unit circle except the point $z=1$, well at $z=1$ we get our known divergent harmonic series, but I am not able to show easily ...
1
vote
2answers
39 views
Divergence of $\sum_{n=2}^\infty\frac{1}{(\ln n)^x}$ for $x>1$
How can we show that the series
$$\sum_{n=2}^\infty\frac{1}{(\ln n)^x}$$
diverges for $x>1$ ?
The book gives the following hint:
consider
$$\sum_{k=1}^\infty\sum_{n=M_{k-1}+1}^{M_k}\frac{1}{(\ln ...
3
votes
3answers
45 views
Value of series, Partialsum?
given is the following series
$$\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$$
And I need to find its value.
How can I start finding it?
Thanks for all
2
votes
1answer
30 views
Monotonically decreasing sequence $(a_n) \to$ mo. dec. sequ. $(a_1+a_2+\cdots+a_n)/n$
I think this isn't quite difficult, however I don't get the point..
I have to prove:
$(a_n)$ is a monotonically decreasing sequence. Show, that the sequence $\frac{a_1 + a_2 + \cdots + a_n}{n}$ is ...
0
votes
1answer
34 views
Relation between a sum of a series and the limit of a sequence
I'm stuck on this question
Let $\{a_{n}\}$ a sequence of real numbers
I need to show the series $\sum_{n=1}^{\infty}(a_{n} - a_{n-1})$ and the sequence $\{a_{n}\}$ are the same nature (convergent ...
3
votes
2answers
44 views
Soving Recurrence Relation
I have this relation
$u_{n+1}=\frac{1}{3}u_{n} + 4$
and I need to express the general term $u_{n}$ in terms of $n$ and $u_{0}$.
With partial sums I found this relation
$u_{n}=\frac{1}{3^n}u_{0} + ...
3
votes
2answers
65 views
limit of series exponential
Compute the limit of the series $$\sum\limits_{n=4}^\infty 3\frac{2^{n+1}}{5^{n-2}}$$
How do you approach these types of problems?
I'm thinking that this one is in indeterminate form, is that ...
-3
votes
0answers
60 views
prove the numbers in a sequence are all square number
Given the sequence $\{y_n\}$ defined by:
$$\begin{align*}
&x_{n+1}=23x_n+2+y_n+2\\
&y_{n+1}=551x_n+24y_n+64
\end{align*}$$
for $n\in\Bbb N$ and $x_1=1,y_1=39$, prove that $x_n$ is a square ...
0
votes
1answer
50 views
convergence of series question
how do you determine if a series converges or diverges? Do you just look at their behavior?
1
vote
3answers
39 views
limits of sequences exponential and factorial
Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences:
(a) $a_n=e^{5\cos((\pi/6)^n)}$
(b) $a_n=\frac{n!}{n^n}$
For part (a) do I just take the limit of the exponent part and ...
1
vote
4answers
34 views
Evaluation of a complex numbers partial sum
Let $w = e^{i\frac{2\pi}5}$. I would like to evaluate
$$w^0 + w^1 + w^2 + w^3 +...+ w^{49}$$
Can anyone please give me an idea how to evaluate the expression?
Thanks in advance
3
votes
0answers
78 views
Compute limit of the sequence $x_n$
Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq ...
3
votes
1answer
32 views
Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$
I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
-2
votes
0answers
39 views
Solve $ z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots $.
I am trying to solve $z\in \mathbb{C}$ in terms of $a\in \mathbb{C}$, where
$$
z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots.
$$
I plugged $z= \sum_{k=0}^\infty c_k a^k $ into the ...
1
vote
2answers
47 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 $$ ...
2
votes
0answers
35 views
Simplify series of exponentials
I would like to simplify the following series:
1.$$\sum_{n=1,odd}^{\infty}\frac{\exp(-an^2)}{(b-n^2)^2}$$
2.$$\sum_{n=2,even}^{\infty}\frac{\exp(-an^2)}{(b-n^2)^2}$$
with $a$ and $b$ $\in ...
2
votes
3answers
59 views
Summation of a finite series
Let
$$f(n) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+...+ \frac{1}{2^n-1} \forall \ n \ \epsilon \ \mathbb{N} $$
If it cannot be summed , are there any approximations to the series ?
8
votes
4answers
114 views
Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$
For $a,b,c>0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$
how can I find the limit of sequence above?
Provide me a hint or full solution.
thanks ^^
0
votes
0answers
43 views
Uniform convergence of $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$.
Given that the series $\sum_{n=2}^{\infty}\dfrac{\mathrm{ln}(n)^{p}}{n}$ is convergent iff $p<-1$.
Show that $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$ is uniform convergent as $x \in ...
2
votes
0answers
26 views
Growth of partial sums of a divergent series
I am trying to find how the product $\prod_{n=0}^Me_n$ grows with $M$, when $$e_n=1+\frac{a_1}{t+n}+\frac{a_2}{(t+n)^2}+\dots$$
with large $t$. So as a first estimate I took $e_n=1+\frac{a_1}{t+n}$ so
...
0
votes
1answer
54 views
Compute limit of the sequence given by $x_1=1$, $x_{n+1}^2=\frac{x_n+3}{2}$
Let $\left(x_n\right)$ be a real sequence such that $x_1=1,x_{n+1}^2=\dfrac{x_n+3}{2},\forall n\geq 1$.Compute $\lim_{n\to+\infty} 3^n\sqrt{\dfrac{9}{4}-x_n^2}$
1
vote
1answer
63 views
3rd grade exercise: “make your own turning pattern”
My 8 year old has been given a worksheet of numeric sequences, e.g. "what are the next numbers in the sequence 11, 12, 14, 15, 17, ..." and "make your own number pattern" and "make your own colour ...
10
votes
1answer
120 views
Uniformly convergence?
I'm not sure wether or not the following sum uniformly converge on $\mathbb{R}$ :
$$\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$$
Can someone help me with it? (I can't use Dirichlet' ...
0
votes
3answers
34 views
Arithmetic progression where the sum of the first $p$ terms is $q$, and the first $q$ terms is $p$
The sum of first $p$ terms of an arithmetic progression (A.P) is $q$, and the sum of the first $q$ terms of the same A.P. is $p$.
Find the sum of the first $p+q$ terms of the A.P.
1
vote
0answers
45 views
How prove this $\sum_{n=1}^{\infty}\dfrac{\zeta_{2}}{n^4}=\zeta^2(3)-\dfrac{1}{3}\zeta(6)$
show that
$$\sum_{n=1}^{\infty}\dfrac{\zeta_{2}}{n^4}=\zeta^2(3)-\dfrac{1}{3}\zeta(6)$$
where
$$\zeta_{m}=\sum_{k=1}^{n}\dfrac{1}{k^m},\zeta(m)=\sum_{k=1}^{\infty}\dfrac{1}{k^m}$$
is true？
because ...
2
votes
1answer
64 views
Bounded sequence in Hilbert space contains weak convergent subsequence
In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence.
Is there any short proof? Thanks a lot.
12
votes
3answers
302 views
$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number
I need help to prove the following result.
$\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
1
vote
2answers
62 views
Summation involving subfactorial function
Inspired by this post:
Does the following series converge; if so, to what value does it converge? $$ \sum_{n = 2}^\infty \left|\frac{!n \cdot e}{n!} - 1\right|$$ I am looking for a closed form for ...
2
votes
1answer
43 views
Infinite Series Problem Using Residues [duplicate]
Show that $$\sum_{n=0}^{\infty}\frac{1}{n^2+a^2}=\frac{\pi}{2a}\coth\pi a+\frac{1}{2a^2}, a>0$$
I know I must use summation theorem and I calculated the residue which is:
...
0
votes
1answer
34 views
How to derive the sum of an arithmetic sequence?
I'm attempting to derive a formula for the sum of all elements of an arithmetic series, given the first term, the limiting term (the number that no number in the sequence is higher than), and the ...
4
votes
1answer
109 views
Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$
By integral test, it is easy to see that
$$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$
converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$]
I am ...
1
vote
2answers
44 views
Prove the convergence of the sequence.
Prove the convergence of the following sequence:
$$x_1 = \sqrt{a}$$
$$x_{n+1} = \sqrt{a + x_n}$$
3
votes
3answers
45 views
What is the sum of this infinite series? Which one is it, Taylors? Binomial?
I am trying to figure which formula to use for this one.
$$\displaystyle\sum\limits_{x=-1}^{-\infty} -x(1-y)p(1-p)^{-x}+\sum\limits_{x=1}^\infty xyp(1-p)^x$$
where $0<y<1$, and $0<p<1$.
...
1
vote
3answers
56 views
Infinite Series question [duplicate]
The first one, the effective resistance is $2R$, then $5R/3$ then $13R/8$ etc....
My job is to find the pattern/equation so I can find the total resistance when $20$ resistors are connected. Of ...
1
vote
2answers
62 views
convergence of series with absolute value
prove or show false:
if $\sum_{n=1}^{\infty}\left |a_{n} \right |$ converges, then $\sum_{n=1}^{\infty}\frac{n+1}{n}a_{n}$ converges as well.
Thank you very much in advance,
Yaron.
0
votes
0answers
28 views
analysis: limit of product of sequences [duplicate]
I would really appreciate help with this question:
Show that if the limit $\lim_{n \to \infty} {a_n} = 0$ and sequence $\{b_n\}$ is bounded, then $\lim_{n\to \infty} a_nb_n =0$
thanks
0
votes
1answer
31 views
what does “in wide sense” mean?
I came across the statement "the sequence increases(in wide sense)".
So my doubt is what does author mean by wide sense?I came across this in number theory book
4
votes
2answers
89 views
Why doesn't the Weierstrass approximation theorem imply that every continuous function can be written as a power series?
I hope that my question in the title is well formulated.
I am a little bit confused with the next exercise from a book:
Argue that there exists functions $f \in C[0, \frac{1}{2}]$ that cannot be ...
1
vote
1answer
29 views
series convergence
i ran into this question:
prove or show false:
if $\sum_{n=1}^{\infty}a_{n}$ is a converging series, but the series $\sum_{n=1}^{\infty}a_{n}^2$ diverges, then $\sum_{n=1}^{\infty}a_{n}$ is ...
7
votes
1answer
72 views
methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$
As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which ...
