Recurrence relations, convergence tests, identifying sequences
0
votes
1answer
32 views
complex holomorphic function which only has finite roots
Suppose that D is a bounded region, f $\in$ H(D)$\bigcap$C($\bar D$).Prove that f has only finite roots if f$\neq$0 on $\partial D$.
5
votes
1answer
77 views
Is zero a limit point of the sequence $(\sqrt n \sin n)$
Is zero a limit point of the sequence $(\sqrt n \sin n)_n$?
6
votes
4answers
110 views
Sum of kth roots ($\sum\sqrt[k]{m}$)
I'm trying to find an asymptotic to $$S(n) = \sum_{k=1}^n\sqrt[k]{m}$$ From computational tests, it seems to grow nearly as slowly as $n$. However even $$\sum_{k=1}^\infty\sqrt[k]{m}-1$$ diverges (for ...
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} = ...
7
votes
1answer
88 views
Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
I need help with calculating this sum:
$$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
1
vote
1answer
39 views
Value of coefficients of the power series when radius of convergence is “less than 1” and “greater than or equal to 1”
Let $\sum_{n=1}^\infty c_n (x-a)^n $ be a power series. As "n" approaches infinity,the value of the coefficients "$ c_n $" may or may not be 0 when Radius of convergence R is such that 0< R ...
2
votes
2answers
53 views
A suspicious way to conclude convergence
Let
$$S \triangleq \sum_{k=0}^\infty\,(-1)^k\quad.$$
On one hand, the sequence of partial sums alternates between $0$ and $1$, therefore, does not get arbitrarily close to any value, and $S$ can't ...
3
votes
1answer
44 views
Show that the following product equals 1 (involves trig)
How can I show that:
$$\prod_{k=1}^{n}\left ( 1+2\cos\frac{2\pi .3^{k}}{3^{n}+1} \right )=1$$
Could you please explain to me how to approach this problem?
Thank you.
3
votes
2answers
123 views
Proving a relation between $\sum\frac{1}{(2n-1)^2}$ and $\sum \frac{1}{n^2}$
I ran into this question:
Prove that:
$$\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}=\frac{3}{4}\sum_{n=1}^{\infty}\frac{1}{n^2}$$
Thank you very much in advance.
6
votes
1answer
44 views
Exercise on convergent series
I am stumped by the following exercise (3.24 in Biler--Witkowski's book "Problems in mathematical analysis"):
Let $f$ be a continuous, increasing function from $[0,+\infty]$ to itself. Show that ...
2
votes
2answers
38 views
power series and sequence
Let $\{ $a_n$ : n\geq 1\}$ be a sequence of real numbers such that radius of convergence of the power series : P(t) = $ \sum\limits_{n=0}^\infty a_n t^n $ satisfies $R > 0$.Then $ a_n \rightarrow ...
0
votes
0answers
72 views
Simplifying $\frac{1}{n}\sum_{k=1}^n f(\frac{1}{k})$
Suppose that $$\displaystyle \forall x\in \mathbb{R}_+^* \quad f(x)=\frac{x^2-1}{4}-\frac{\ln(x)}{2}.$$
How can I simplify this: $$I(n)=\frac{1}{n}\sum_{k=1}^n f\left(\frac{1}{k}\right)$$
and prove ...
2
votes
1answer
34 views
Hermite's equation of order $\alpha$
Show that the general solution of Hermite's equation of order $\alpha$:
$${y}''-2x{y}'+2\alpha y=0$$
$$is$$
$$y(x)=c_{0}y_{1}(x)+c_{1}y_{2}(x)$$
where $y_{1}(x)$ and $y_{2}(x)$ are power series ...
2
votes
2answers
29 views
Proving the coefficient of Power series is “0” always on given condition.
Suppose the power series $P(x) = \sum_{n=1}^\infty b_n x^n$ converges for $|x| \leq 1$ and that for some $c>0$ it is given that $$P(x)=0 \quad \forall x \;\text{such as}\;|x| < c$$ Show that ...
3
votes
1answer
47 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.
41
votes
3answers
578 views
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.
I found a solution by myself 10 hours after I posted it, here it is:
...
1
vote
3answers
30 views
Proof that $\sum_{k=0}^m \binom{m}{k}\frac{1}{k+1} = \frac{2^{m+1}-1}{m+1}$ [duplicate]
Recently I needed to compute $E[\frac{1}{X+1}]$ where $X\sim Bin(m, \frac 1 2)$.
While expanding, I came across the sum $\sum_{k=0}^m \binom{m}{k}\frac{1}{k+1}$, which I was unable to solve. Plugging ...
0
votes
0answers
38 views
A question with infinity [Part 2]
If you haven't seen my first post regarding infinity you can find it here: A question with infinity
Thanks for all the constructive comments on my first post, and creative answers to my questions.
...
4
votes
3answers
59 views
Infinite product involving powers of 2
I have the following infinite product:
$2/1 * 3/2 * 5/4 * 9/8 * 17/16 * 33/32 * 65/64...$
What does it converge to?
I can take its $\ln()$ to get
$\ln(2) + \ln(3/2) + \ln(5/4)....$
Which using ...
1
vote
2answers
40 views
Finding the $x^n$ coefficient of the power series $\sum\limits_{n=0}^\infty\frac{x^{2n+3}}{n!}$
I have a practice test question that asks:
Given the following Maclaurin series representation, $$\sum\limits_{n=0}^\infty\frac{x^{2n+3}}{n!}$$ what is the coefficient of $x^n$?
I have the ...
0
votes
2answers
67 views
A few problems on sup and nested intervals
I've been doing these 3 problems for a `proof´ oriented class, one i have found a solution (in fact has been asked here before but the threads are all closed), and checked a correct solution in the ...
6
votes
4answers
110 views
$\sum_{n=0}^{\infty}(-1)^n a_n = \pi$, $a_n\in \mathbb Q$ and not $a_n$ not monotonic
How can I construct a sequence $\{a_n\}$ of positive rational numbers, which is not monotonic such that $$\sum_{n=0}^{\infty}(-1)^n a_n = \pi$$
I thought a lot on this question but I always come up ...
2
votes
1answer
38 views
Pick the highest of two (or $n$) independent uniformly distributed random numbers - average value?
With "random number" I mean an independent uniformly distributed random number.
One
Picking one random number is easy: When I pick a random number from $x$ to $y$, the average value is $(x+y)/2$.
...
7
votes
1answer
48 views
Divergent series of positive numbers
So I've been trying to figure out how to prove the following.
Let $(a_n)$ be a sequence of positive numbers such that $\sum_{n=1}^\infty a_n =\infty$, and define $s_n=\sum_{i=1}^n a_n$. Then ...
-1
votes
0answers
64 views
Treating indices as if they were exponents [closed]
Suppose $G(x) = a_0 x^0 + a_1 x^1 + a_2 x^2 + a_3 x^3 + \cdots$ and I wish to write
$$a_0 x^0 + a_1 x^1 + a_2 x^2 + a_3 x^3 + \cdots \equiv a^0 x^0 + a^1 x^1 + a^2 x^2 + a^3 x^3 + \cdots$$
How can I ...
8
votes
0answers
119 views
A series: $1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\cdots$
Denote $$b_1=1,b_{n}=b_{n-1}-\dfrac{S(b_{n-1})}{n},(n>1 )\tag1$$
where $S(x)=1$ if $x>0,S(x)=-1$ if $x<0$, and $S(0)=0.$
So ...
2
votes
1answer
61 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 $U_n$ ...
1
vote
1answer
65 views
$\lim_{n \to \infty}\sum_1^n \frac{(\log k)^4}{ k^2}$ converges?
Does the series converge?
$$\lim_{n \to \infty}\sum_1^n \dfrac{(\log k)^4}{ k^2}$$
3
votes
1answer
90 views
Does there exist such a sequence?
Does there exist a infinite sequence of positive integers $a_n$. Such that $ ((n|a_n) | \forall n)$ and $\left(\sum_{n=1}^\infty \frac {1}{a_n} =1\right)$ , what if we replace 1 with a positive real ...
1
vote
2answers
47 views
is $\sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right)<\infty$?
I don't know why but I'm having a hard time determining whether this series
$$
\sum\limits _{n=1}^{\infty}\ln\left(\frac{\left(n+1\right)^{2}}{n\left(n+2\right)}\right)
$$
converges to a real limit.
...
3
votes
3answers
30 views
A problem on recurrence relation
Consider the sequence $$a_n = a_{n-1} a_{n-2} +n$$ for $n \geq 2$, with $a_0 = 1$ and $a_1 = 1$. Is $a_{2011}$ odd?
By writing all the terms of the sequence I see that $a_n$ is odd when $n$ is odd ...
2
votes
3answers
107 views
Derivative of a ratio of geometric series
I am trying to prove a theorem in my paper and am stuck at this irritating thing. Please help me.
Show that $$\frac{d}{dk}\left(\frac{\sum_{x=1}^{n} x*k^x}{\sum_{x=1}^{n} k^x}\right) > 0$$ where $n ...
0
votes
1answer
59 views
Differentiable functions and sequences
Let $f(x)$ be a differentiable function on $R$ with $\left|\,f'(x)\right|\leq r < 1$, here $r$ is a constant. Consider the sequence $\{x_n\}$ such that $x_1=0$, $x_{n+1}=f(x_n)$, $n\geq 1.$ Show ...
2
votes
3answers
101 views
Evaluate the infinite series $\sum_{r=1}^{\infty} \frac{2^{-r}}{r^{2}}$
My teacher posed an infinite series question to me today and I'm not quite sure how to start to go about it.
$$\sum_{r=1}^{\infty} \dfrac{2^{-r}}{r^{2}}$$
Any hints would be much appreciated.
1
vote
3answers
29 views
Find Simple Convergent sum
We have a sum
$\sum_{0}^{\infty} 2^{n+1}(x+1)^{3n+1}$
I am asked to find all values of x s.t this sum converges then compute the sum.
I have no idea what im doing so i did this. $\frac ...
1
vote
0answers
26 views
Is this series of hermite polynomials $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} H_n(x)$ a known function?
Is this series of hermite polynomials $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} H_n(x)$ a known function?
Thank you!
0
votes
1answer
24 views
Determining Fourier series for $\lvert \sin{x}\rvert$ for building sums
My math problem is a bit more tricky than it sounds in the caption.
I have the following Task (which i in fact do not understand):
"Determine the Fourier series for $f(x)=\lvert \sin{x}\rvert$ in ...
3
votes
3answers
47 views
Problem related to a series
I am stuck on the following problem:
Let $\{p_n\}$ be the sequence of consecutive positive solutions of the equation $\tan x=x$ and let $\{q_n\}$ be the sequence of consecutive positive ...
2
votes
3answers
44 views
Finding the explicit formula for a recursive sequence, using power series
The Task is to find the explicit expression for the given recursive sequence with the help of power series.
Given:
$a_{0}=0,\ a_{1}=1 \quad$ and $\quad a_{n}=5\cdot a_{n-1} -6\cdot a_{n-2}\quad $ ...
0
votes
0answers
23 views
Construct a sequence with good property
I want to construct a sequence, the question is following:
Given $1<\beta_1<\beta_2<\beta_3<\cdots<\beta_M<2$, then construct a sequence $(h_1\,,h_2\,,\cdots\,,h_L)$ where $L$ is ...
12
votes
2answers
119 views
Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$
How am I supposed to prove this inequality?
$$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$
Jordan inequality might be an option but led me ...
0
votes
0answers
16 views
Series of two unknown initial variables
\begin{align}S_n&= c_1 S_{n-1}- c_2 (V_n )^2 ,\\
V_n&= V_{n-1}+1/(S_{n-1} )^2
\end{align}
What are the values of $S_0$ and $V_0$ that satisfy that the value of $S_n ≥1$ after any ...
1
vote
3answers
130 views
Evaluate a sum with binomial coefficients
$$\text{Find} \ \ \sum_{k=0}^{n} (-1)^k k \binom{n}{k}^2$$
I expanded the binomial coefficients within the sum and got $$\binom{n}{0}^2 + \binom{n}{1}^2 + \binom{n}{2}^2 + \dots + \binom{n}{n}^2$$
...
2
votes
0answers
34 views
how to prove the limit of a sequence using limit definition?
Please, help me to prove that $x_n=\frac {n^2+n}{2n^2-3n-4}\rightarrow\frac{1}{2}$.
$\forall \epsilon>0, \exists n_0 \in \mathbb N$ such that $n>n_0\Rightarrow$
...
3
votes
2answers
76 views
Power series and the value of the expression $0^0$
I have a doubt regarding the value of the expression $0^0$. I know this value is taken as indeterminate as far as limits are concerned. All was fine upto now. But when I encountered power series, I ...
3
votes
2answers
131 views
Automata: 1=2, 2= 26, 3=1054, 4=5768, 5 =139314069504, 6 = ???
I am in my own Automaton (finite-state deterministic automata) research, so i have four sets of automata.
2 states automata,
3 states,
4 states and
5 states.
Input alphabet $\{0,1\}$
so...
the ...
12
votes
3answers
108 views
Find a number $x$ such that $\sum_{n=1}^\infty\frac{n^x}{2^n n!} = \frac{1539}{64}e^{1/2}$
I need to find a number $x$ such that
$$\sum_{n=1}^\infty\frac{n^x}{2^n n!} = \frac{1539}{64}e^{1/2}.$$
What is the best approach to this problem?
1
vote
4answers
60 views
Sequence Diverging: Where's My Mistake?
EDIT: Question answered; I misread my own handwriting when copying my notes into LaTeX. Thanks!
I'm trying to show that $a_{n}$ diverges. The equation I arrive at does not diverge. Where did I go ...
5
votes
2answers
77 views
Determine the character of $\sum_{n=1}^{+\infty}{\frac{e^{i\theta n}}{n}}$
Determine the character of the following series:
$$\sum_{n=1}^{+\infty}{\frac{e^{i\theta n}}{n}}$$
where $\theta$ is a real parameter.
I try to divide the series with De Moivre' s formula:
...
3
votes
2answers
75 views
Does the series $\sum_{n=1}^{\infty}|x|^\sqrt n$ converge pointwise? If it then what would be the sum?
Does the series $\sum_{n=1}^{\infty}|x|^\sqrt n$ converge pointwise?
