Recurrence relations, convergence tests, identifying sequences
0
votes
0answers
11 views
the relationship between generating function and recurrence
Does the existence of a generating function of a sequence imply the existence of recurrence relation of the same sequence?
0
votes
0answers
17 views
Are these series convergent?
I came across the following two series while trying to solve Laplace's equation in two dimensions.
$$T_1 = \sum_{n=1}^{\infty}\rho^n(A_n\cos(n\phi) + B_n \sin(n\phi))$$
$$T_2 = ...
3
votes
2answers
72 views
$\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$
Let $\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$,$s_k$ denotes the partial sum then could anyone tell me which of the following is/are correct?
$1.$ $s_k=0$ for ...
2
votes
2answers
24 views
Find an interval of convergence and an explicit formula for $f(x)$
Let $f(x) = 1 + 2x + x^2 + 2x^3 +x^4+...$
If $c_{2n} = 1$ and $c_{2n+1} = 2$ $\forall n \ge 0$ find the interval of convergence and an explicit formula for $f(x)$.
The answers are $I = (-1,1)$ and ...
0
votes
0answers
19 views
Mapping Between Sequences: Example
Take $0\leq r < j$, and let all values be nonnegative and integer. Consider the function on a sequence ${x(n)}$, $\Phi_jx(mn+r)=mx(n)+\frac{r}{m}(x(n+1)-x(n))$, where we consider $x(0)=0$.
As an ...
11
votes
2answers
149 views
An infinite product
Given that $0 < a < 1$, what is the value of
$$ P = \lim_{n\to \infty} \prod_{i=1}^n \frac{1 - (2i + 1)a/(2n)}{1 - ia/n} $$
Thus, $P_n$, the $n$th term, is
$$ P_n = \frac{1-3a/2n}{1-a/n}\cdot ...
1
vote
1answer
29 views
Series of Vectors
In $\mathbb{R}^n$ we define sequences of it's elements in a very natural manner, we say that a sequence is a function $x : \mathbb{N} \to \mathbb{R}^n$ and we denote it by $(x_k)$ as in the $n=1$ ...
2
votes
2answers
17 views
Question on AP(sequences and series)
Prove that sqrt(2), sqrt(3) and sqrt(5) cannot be terms of an A.P.(not necessarily consecutive)!
2
votes
1answer
26 views
Calculation Of Integral Related To Sequence
Let's evaluate the following integral. Many trials but no success.
$$\int_{-\pi}^{\pi}\dfrac{\sin nx}{(1+\pi^{x})\sin x}dx$$
0
votes
1answer
27 views
How to represent a sequence of odd numbers given specific criterea
I'm trying to figure out how to represent a sequence of ODD numbers given the following conditions:
1) I know how many numbers are in the sequence (N).
2) I know the average of all the numbers in ...
1
vote
1answer
43 views
Taylor series expansion and approximation
I found this amazing question in the last calculus exam, but I don't know how to answer.
Let $T(x) = \ln(1+a) + \frac{1}{1+a}(x-a) - \frac{1}{2(1+a)^2}(x-a)^2 +...+ ...
7
votes
1answer
91 views
convergence of series with $k!$
check if the following series converges:
$\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$
where $k!!=k(k-2)(k-4)(k-6)...$
I came across this exercise while going trough some old exams. I'm ...
1
vote
1answer
27 views
identity of polylogarithm
let be the function defined by a series
$$ f(x)= \sum_{n=0}^{\infty}g(n)x^{n} $$
assume also that $ g(n)= \sum_{k=0}^{\infty}a(k)n^{k} $
then we have the double series
$$ f(x)= ...
2
votes
3answers
97 views
Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$
Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that
$$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$
?
I think it is but I can't prove it. Of course if $a_n ...
1
vote
0answers
28 views
expanded geometric series?
I'm having some issues with the following series
$$
\sum_{n \geq 0} n^p r^n
$$
for a fixed positive integer $p$ and some real $r > 0$.
Is there any way to avoid going through linear combinations ...
0
votes
0answers
25 views
The generating function for Bernoulli polynomials
The generating function for Bernoulli polynomials is given by:
$$\frac{ue^{ux}}{e^u-1}=\sum_{n\geq 0}B_n(x)\frac{u^n}{n!}$$
Now, I have the following expression:
...
1
vote
0answers
68 views
Proving that every term of the sequence is an integer
Let $m,n$ be nonnegative integers.
The sequence $\{a_{m,n}\}$ satisfies the following three conditions.
For any $m$, $a_{m,0}=a_{m,1}=1$
For any $n$, $a_{0,n}=1$
For any $m\ge0, n\ge1$, ...
3
votes
3answers
51 views
Integration using summation
How do you integrate $\sqrt{x}$ from an arbitrary constant $a$ to another $b$ by summation ?
0
votes
0answers
22 views
there are different series to evaluate $\pi$ but which of it can evaluate $\pi$ Rapidly and how they prove that series?
there are different way to evaluate $\pi$ by a series but which series can evaluate $\pi$ Rapidly and how to prove that series ?
2
votes
1answer
30 views
Intuition behind closed subsets of a metric space?
Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space.
Consider a metric space $$(X,d)$$
Then consider a subset of this space$$F$$
What the book ...
2
votes
1answer
43 views
Does it make sense to talk about the concatenation of infinite series?
Does a series of numbers defined as the concatenation of two or more infinite series, for example all the positive integers followed by all the negative integers, make mathematical sense?
I came up ...
0
votes
2answers
67 views
Taylor series of $f(x)=\frac {e^x-1}{x}$
I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions.
How to simplify the function so that it can be expanded more easily?
1
vote
1answer
46 views
Math arithmetic
So I got this question in my exam yesterday, and only a few people from my school could solve this.
What's so hard about it? I have learned the simple arithmetic questions, but I don't really ...
2
votes
3answers
39 views
find the condition on A for the summation to be convergent
The summation is:
$$\sum_{n=1}^\infty \frac{ \sqrt { n + 1 } - \sqrt n }{n^A}$$
I don't know how to even begin. Hints??
0
votes
0answers
40 views
funcitonal series convergence… SOS… [duplicate]
Is it possible to show that $f(x) = \sum_{k=0}^\infty (-1)^k \frac{x^k \sqrt{k}}{k!}$ converges even when $x\rightarrow \infty$ ?
i.e. As a function of x, does $\lim_{x\to\infty} f(x)$ exist or at ...
-1
votes
1answer
38 views
Alternating functional Series Convergence SOS…
Does the following series converge?
$\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$
what is the radius of convergence?!!
1
vote
1answer
30 views
How do I write the generic series definition that can produce any term in the series expansion?
I'm trying to learn how to build the generic series definition for a series of numbers. For some reason I'm having a hard time pulling out this pattern. I'm always pulling out the wrong details for ...
1
vote
2answers
34 views
Extracting distinct sequence
Let $(x_n)$ be a non-constant sequence in $\mathbb{R}$ and $x_n\rightarrow p$ for some $p\in \mathbb{R}$. Can I always extract a subsequence $(x_{n_k})$ whose all elements are distinct? (of course ...
16
votes
1answer
160 views
Prove every integer exists in this sequence?
Please could someone give me a hint on this sequences question? The question is to prove that every integer appears infinitely many times in the following sequence:
$$ \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
23 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 $w_t$ ...
2
votes
0answers
31 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
45 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
$$\sum_{k=1}^\infty\sum_{n=M_{k-1}+1}^{M_k}\frac{1}{(\ln ...
3
votes
3answers
46 views
Value of series, Partialsum?
given is the following series
$$\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$$
And I need to find its value.
How can I start finding it?
Thanks for all
2
votes
1answer
31 views
Monotonically decreasing sequence $(a_n) \to$ mo. dec. sequ. $(a_1+a_2+\cdots+a_n)/n$
I think this isn't quite difficult, however I don't get the point..
I have to prove:
$(a_n)$ is a monotonically decreasing sequence. Show, that the sequence $\frac{a_1 + a_2 + \cdots + a_n}{n}$ is ...
0
votes
1answer
35 views
Relation between a sum of a series and the limit of a sequence
I'm stuck on this question
Let $\{a_{n}\}$ a sequence of real numbers
I need to show the series $\sum_{n=1}^{\infty}(a_{n} - a_{n-1})$ and the sequence $\{a_{n}\}$ are the same nature (convergent ...
3
votes
2answers
46 views
Soving Recurrence Relation
I have this relation
$u_{n+1}=\frac{1}{3}u_{n} + 4$
and I need to express the general term $u_{n}$ in terms of $n$ and $u_{0}$.
With partial sums I found this relation
$u_{n}=\frac{1}{3^n}u_{0} + ...
3
votes
2answers
66 views
limit of series exponential
Compute the limit of the series $$\sum\limits_{n=4}^\infty 3\frac{2^{n+1}}{5^{n-2}}$$
How do you approach these types of problems?
I'm thinking that this one is in indeterminate form, is that ...
-4
votes
1answer
68 views
prove the numbers in a sequence are all square number
Given the sequence $\{y_n\}$ defined by:
$$\begin{align*}
&x_{n+1}=23x_n+2+y_n+2\\
&y_{n+1}=551x_n+24y_n+64
\end{align*}$$
for $n\in\Bbb N$ and $x_1=1,y_1=39$, prove that $x_n$ is a square ...
0
votes
1answer
51 views
convergence of series question
how do you determine if a series converges or diverges? Do you just look at their behavior?
1
vote
3answers
39 views
limits of sequences exponential and factorial
Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences:
(a) $a_n=e^{5\cos((\pi/6)^n)}$
(b) $a_n=\frac{n!}{n^n}$
For part (a) do I just take the limit of the exponent part and ...
1
vote
4answers
38 views
Evaluation of a complex numbers partial sum
Let $w = e^{i\frac{2\pi}5}$. I would like to evaluate
$$w^0 + w^1 + w^2 + w^3 +...+ w^{49}$$
Can anyone please give me an idea how to evaluate the expression?
Thanks in advance
3
votes
0answers
81 views
Compute limit of the sequence $x_n$
Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq ...
3
votes
1answer
33 views
Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$
I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
-2
votes
0answers
40 views
Solve $ z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots $.
I am trying to solve $z\in \mathbb{C}$ in terms of $a\in \mathbb{C}$, where
$$
z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots.
$$
I plugged $z= \sum_{k=0}^\infty c_k a^k $ into the ...
1
vote
2answers
48 views
Convergence of these series
$$\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
60 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
130 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 ...
