Recurrence relations, convergence tests, identifying sequences
0
votes
0answers
20 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
29 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
24 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
...
-1
votes
1answer
39 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
55 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 ...
0
votes
3answers
31 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
35 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
53 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.
13
votes
3answers
276 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
43 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
40 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
102 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
44 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
55 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
57 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
86 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
28 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
67 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 ...
8
votes
3answers
95 views
Help me prove this inequality :
How would I go about proving this?
$$ \displaystyle\sum_{r=1}^{n} \left( 1 + \dfrac{1}{2r} \right)^{2r} \leq n \displaystyle\sum_{r=0}^{n+1} \displaystyle\binom{n+1}{r} \left( \dfrac{1}{n+1} ...
0
votes
1answer
26 views
proving a z transform
I am having trouble demonstrating the Z transform of $a^{n-1}u(n-1)$ is $\frac{1}{(z-a)}$, as it says in this table.
I try using the definition of the z transform, but it comes out different than ...
1
vote
0answers
40 views
Maclaurin series expansion of an expression that involves a fraction
In the context of statistical mechanics the "classical trace" is defined as $Tr(A e^{-\beta H}) = \int dr^N dp^N A e^{-\beta H}$ where $r^N$ and $p^N$ are phase space variables. So if $\Delta H$ is a ...
2
votes
1answer
20 views
Sequence of continuous functions, integral, series convergence
Let $f_k$ be a sequence of continuous functions on $[0,1]$ such that $\int _0 ^1 f_k(x)x^ndx = \int _0^1 x^{n+k} dx$ for all $n \in \mathbb{N}$.
Is $\sum _{k=1} ^{\infty}f_k(x)$ convergent?
Could ...
1
vote
0answers
25 views
What is the broader name for fibonacci and lucas sequences?
Fibonacci and Lucas sequences are very similar in their definition. However, I could just as easily make another series with a similar definition; an example would be:
$$x_0 = 53$$
$$x_1 = 62$$
$$x_n ...
1
vote
2answers
32 views
Is there any specific terminology to refer to an initial sequence of a sequence?
Lets say you have a sequence $S = (0, 1, 2, 3, 4, 5, 6, 7, 8)$
And another sequence $T = (0, 1, 2, 3)$
Is there any specific mathematical term that defines the relationship between $S$ and $T$, ...
3
votes
4answers
63 views
Limit of a recursive sequence with $u_n$
It is given that $u_{n+1} =1+\dfrac{1}{u_n}$ and $u_1 =1$. Find the limit of $u_n$ as $n\to\infty$.
The limit is $\frac{\sqrt{5}+1}{2}$ from a calculator. Is there an algebraic way to determine this? ...
0
votes
2answers
55 views
Sequence $(a_n)$ s.t $\sum\sqrt{a_na_{n+1}}<\infty$ but $\sum a_n=\infty$
I am looking for a positive sequence $(a_n)_{n=1}^{\infty}$ such that $\sum_{n=1}^{\infty}\sqrt{a_na_{n+1}}<\infty$ but $\sum_{n=1}^{\infty} a_n>\infty$.
Thank you very much.
2
votes
3answers
70 views
If $\sum a_n $ is a positive series that diverges, does $\sum \frac{a_n}{1+a_n}$ diverge? [duplicate]
Let $ \{a_n\}_{n=1}^\infty $ be a sequence such that $\displaystyle \sum a_n $ that is divergent to $+\infty$.
What can be said about the convergence of $\displaystyle \sum \frac{a_n}{1 + a_n} $?
...
2
votes
2answers
49 views
alternating series test for $\sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}$
I must determine if this series converges (using specifically the alternating series test) $$\sum_{n=4}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}.$$
I know the necessary and sufficient conditions are:
The ...
4
votes
2answers
73 views
Closed form for $\sum^{\infty}_{{i=n}}ix^{i-1}$
How can I find a closed form for:
$$\sum^{\infty}_{{i=n}}ix^{i-1}$$
It looks like that's something to do with the derivative
1
vote
3answers
50 views
On convergence of problematic series.
Determine if the following series is converges or not
$$\sum_{n=2}^{\infty}\frac{1}{\ln^{\ln(\ln n)} (n)}$$
3
votes
1answer
40 views
5
votes
2answers
48 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
7
votes
2answers
79 views
If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
I'm going over old exam problems and I got stuck on this one.
Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be
the non-zero zeros of ...
2
votes
1answer
34 views
Convergence of sequence
Does the following:
$$
\begin{align}
x_0 & = a \\
x_1 & = x_0 + \frac{1}{1}(x_0 + x_0(c - 1)) \\
x_2 & = x_1 + \frac{1}{2}(b + x_1(c - 1)) \\
x_3 & = x_2 + \frac{1}{3}(b + x_2(c - 1)) ...
1
vote
3answers
54 views
Showing that $\sum_{n=3}^\infty\frac1{n(\log n)(\log\log n)}$ diverges
I know that the series diverge, I'm just having hard time showing it.
$$\sum_{n=3}^\infty\frac1{n(\log n)(\log\log n)}$$
Thanks in advance
3
votes
2answers
37 views
Is a sequence of all the same numbers monotonic?
I'm wondering based on the definition of monotonicity:
A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic.
So given that the sequence $a_n = 3$ is all the same numbers and ...
2
votes
3answers
58 views
Series Summation
I have the series
$$\sum_{k=1}^{N-1}\frac{1}{1-w_k} $$ where $w_k=e^{\frac{2\pi i k}{N}}$, how can I find the summation of this , another question related to this sum
$$\sum_{k=1}^{N-1}\frac{1}{z-w_k} ...
0
votes
2answers
27 views
Series expansion with remaining $log n$
I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant.
$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$
I'm trying to do a series ...
1
vote
2answers
55 views
Partial fraction expansion two variables
How to expand
$$\frac{y}{(x-y)(y-1)}$$
by partial fraction expansion.
6
votes
2answers
72 views
Evaluate the Sum $\sum_{i=0}^\infty \frac {i^N} {4^i}$ [duplicate]
$$\displaystyle\sum_{i=0}^\infty \frac {i^N} {4^i}$$
I'm supposed to evaluate this as I'm working through Data Structures and Algorithm Analysis in C++. I've solved similar problems, and after ...
1
vote
0answers
39 views
Bounding a sequence defined recursively
Let $\{x_n\}$ be a sequence of positive numbers and $\alpha \in (0,1)$. Let $y_0 := x_n$ and
$$
y_k := (\alpha \,y_{k-1} + 1-\alpha) x_{n-k}
$$
for $k=1,2,\dots,n-1$.
Is it possible to give a sharp ...
0
votes
0answers
38 views
Linear Combinations of Irrational Numbers: An Analysis on Architecture
Under what condition(s) is
$$ k_1\omega_1+\cdots + k_n\omega_n=c,$$
where $k_i\in\mathbb{R\setminus Q}$ and $\omega_i, c\in \mathbb{Q}$?
I'm essentially trying to show that this is the case only so ...
1
vote
6answers
114 views
Does $\sum_{n=1}^\infty(-1)^{n}\frac{n^n}{n!}$ converge or diverge
$\sum_{n=1}^\infty(-1)^{n}\frac{n^n}{n!} $
I got that it diverges but I am not sure
0
votes
1answer
53 views
Sequence version of L'Hospital's Rule
Consider two sequences $A_n$ and $B_n$ such that $B_n$ is monotonically decreasing and both $A_n$ and $B_n$ tend to zero.
Now let us consider the limits ...
0
votes
1answer
39 views
What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$
For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series
$$
\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n.
$$
For $b=0$, this post shows
$$
\sum_{n=0}^\infty ...
0
votes
1answer
29 views
Taylor and geometric series
1) iF $f(x) = x^2 +x$, find the taylor series for f centered at a = 2.
2)what is the sum from 1 to infinity of $(.95)^n$
I got these questions wrong on my last test, and I'm not really sure how to ...
