Tagged Questions
2
votes
2answers
30 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 ...
4
votes
3answers
48 views
$f(x)=\tanh(1+\tanh^{-1}(x))$ or $f:\tanh(x) \to \tanh(x+1)$ is a rational function?
This is (again) more a recreational/incidental question.
Playing with iteration of functions I considered the function $$ f(x) = \tanh(1+\tanh^{-1}(x)) \tag1$$ such that $$ f : \tanh(x) \to ...
0
votes
1answer
77 views
Prove this proprety of $f(x)$
I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$?
Let ...
0
votes
2answers
39 views
Show uniform convergence of indefinite function series
How can i show uniform convergence on function series like this one: $\sum\limits_{k=1}^{\infty} (\sqrt{1-x^{n}}-1)$ ? I have a given interval of [0 / 0.5]
I thought about using the Weierstrass ...
8
votes
1answer
153 views
Function mapping challange
For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant
mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$
and $f(k) = f(f(k + 1))$.
This is ...
1
vote
1answer
56 views
nth term test for divergence - help
$$\sum_{n=1}^{\infty} \left(\dfrac{n}{n+1}\right)^n$$
to show that this diverges should I use the $n^{th}$ term test?
So far I have substituted infinity for $n$. Could I use L'hopital's rule to ...
1
vote
1answer
20 views
How do we determine as to how long we should sum an asymptotic series of a function to get the answer correct up to a particular precision?
As an example, consider the asymptotic expansion for polygamma function . What should be the min value of 'k' in the equation to get the answer correct upto a particular precision, say pth. Is there ...
2
votes
1answer
20 views
Proof Regarding Series of Functions.
If it's not too much trouble, may I have some help on this question regarding series of functions?
Let $u$ and $v$ be two series of functions on a set $X$ such that $|u| < |v|$ for every $x$ ...
1
vote
2answers
55 views
$f(x) = \frac {e^{2x-1}} {(1+e^{2x-1})}. $ What is the value of $ f(1/2009) + f(2/2009) + … + f(2008/2009) $?
Here's the question.
Let $f(x) = \frac {e^{2x-1}} {(1+e^{2x-1})}$
Then what is the value of $ f(1/2009) + f(2/2009) + ... + f(2008/2009) $ ?
All I could think of doing was to add and subtract 1 in ...
0
votes
2answers
39 views
Function of series
If a function $F(x) = \sum_i a_i/(x - y_i)$. Is it possible to simplify F(x) so that the repeated sum for each value of x can be avoided ?
5
votes
0answers
229 views
Extension of the Jacobi triple product identity
The Jacobi triple product identity is:
$$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$
I would like to extend the idea for ...
0
votes
2answers
43 views
Proof of functional equation
I have a function
$$ g_N(x) =\frac{1}{x}+ \sum_{n=1}^\infty \Big(\frac{1}{x+n}+\frac{1}{x-n}\Big) $$
How can I prove that $$g_N(\frac{x}{2})+g_N(\frac{x+1}{2}) = 2g_N(x)\;?$$
3
votes
1answer
138 views
Power Series representation of $\frac{1+x}{(1-x)^2}$
Can anyone work out how to do this problem, because I'm getting an answer that close to the answer in the back of the book, but mine is off by a + 1.
What I do is, I first find a representation for ...
1
vote
1answer
57 views
Sequence of continuous functions converges uniformly. Does it imply the limit function is continuous?
Let $f_n \in C[a; b]$, where $\{f_n\}$ converges uniformly to $f$.
Is it true that $f \in C[a; b]$ too ? How do I prove or disprove it?
1
vote
0answers
54 views
How to analyze limit of function sequences?
While generally analyze the functional series so there is the ultimate techniques how to deal with them. There is square root criterion, unit test, comparison with $ \frac{1}{a_n ^ q} $ and limit ...
0
votes
3answers
99 views
Find the function that satisfies the following
Let $f: \mathbb{R} \to \mathbb{R}$ inconstant so that $\exists \lim_{x \to +\infty} f(x) $ and for any arithmetical progression $(a_n)$ the sequence $(f(a_n))$ is an arithmetical progression.
...
0
votes
2answers
66 views
Lipschitz Functions and Cauchy Sequences
I need help with a particular question:
A function $f(x)$ is Lipschitz on $\mathbb R$ if $\exists C$ that is a constant such that $\forall x, y \in \mathbb R,|f(x)-f(y)| \leq C|x-y|$.
Show that if ...
2
votes
1answer
127 views
Find power series representation for $f(x) = (4 − x)^{−3}$
Find a power series representation centered at the origin for the function $$f(x) = (4 − x)^{−3}.$$
1
vote
1answer
148 views
Lim Sup/Inf for real valued functions
To understand the notion of, say, limit superior for a sequence, is not difficult. Simply consider the set of all upper buonds for the set of all limit points of the sequence, and then simply pick the ...
4
votes
3answers
99 views
Bounds for solutions of $xe^{x} -n =0$ for $n\geq 3$
$\alpha _{ n }$ is a solution for $f_{ n }=xe^{ x }-n =0$. How can I prove that $\forall n\geq 3$, $\ln(n)-\ln(\ln(n))\le \alpha _{ n }\le \ln(n)$
0
votes
1answer
99 views
Can a sequence of functions converge to different functions pointwise and on average?
Is it possible for a sequence of functions in $\mathcal{C}\left[0,1\right]$
to converge to one function pointwise (not necessarily to a continuous function) and to a different function in average ...
3
votes
2answers
125 views
Infinite sum of floor functions
I need to compute this (convergent) sum
$$\sum_{j=0}^\infty\left(j-2^k\left\lfloor\frac{j}{2^k}\right\rfloor\right)(1-\alpha)^j\alpha$$
But I have no idea how to get rid of the floor thing. I thought ...
6
votes
2answers
141 views
Currious property of monotonic functions
If $f(x)$ is continuous and monontonicly increaseing on the interval $[1,\infty]$, and $f'(x)\leq\frac{1}{x}$ on the interval $[1,\infty]$,
is $$\lim_{n \rightarrow \infty} ...
0
votes
1answer
97 views
Let $f$ be a real valued sequentially continuous function relative to a closed bounded interval $I=[a,b]$. Prove that the set $f(I)$ is bounded above
The hint that I've been given is: for each n in the naturals, use the assumption that $n$ is not an upper bound for $f(I)$ to choose a sequence of $x_n$ (from $n=1$ to infinity) in $I$; then apply ...
1
vote
2answers
77 views
Does this series approach an actual value?
Does this series converge to an actual number for any value of $x$? $$\sum_ {k = 1}^{\infty} \frac{\ln(k)\sin( 2\pi kx) }{ k}. $$ I tried summing the series for $x=2/3$ on wolfram alpha, and it seems ...
1
vote
2answers
58 views
Periodic series help
Is there an infinite series composed possibly of periodic functions for a function $f(k)$ with the property that if, $ k\equiv b $ mod a, $f(k)=1$, and if not $f(k)=0$,
2
votes
1answer
60 views
Simple function help
I am not familiar with Fourier series, (I'm guessing that has something to do with what I want), and I want to know if someone could construct a convergent series for a function $f(x)$ with the ...
2
votes
2answers
49 views
Study of a series of functions
I've to study this series:
$$\sum_{n=1}^\infty e^{\sqrt n\,x}$$
My teacher wrote that with the asymptotic comparison with this series:
$$\sum_{n=1}^\infty\frac{1}{n^2}$$
My series converges ...
5
votes
2answers
83 views
Do zeros of uniformly convergent function sequences also converge?
Assume the following:
$f_n{(x)}$ is a sequence of continuous functions, each with a unique zero $x_n^*$
$f_n\to f$ uniformly
$f$ has a unique zero at $x$
Does it then follow that $x_n^*\to x$?
If ...
1
vote
2answers
133 views
Proving Functions are Surjective
Prove the following are surjective, or disprove with a counter-example:
$f\colon \mathbb{Q} \to \mathbb{Q}$, $f(x) = 1 + 2x$.
$f\colon \mathbb{Z} \to \mathbb{N}\cup\{0\}$, $f(x) = |1 - x|$.
...
0
votes
2answers
47 views
Would cartesian product be the best approach for this
Not sure on how to migrate a question yet but over on SO someone said I might get better results here. Also please retag as I'm not allowed to create new and might not know the best tagging.
Link to ...
1
vote
0answers
23 views
Series function help
I want to find a function such that $$ \sum_{0<j<n/k
} f(kj)=1 $$
Where the sum j is taken over the natural numbers,
And the series is satisfied for all integers k and n, I was thinking of ...
1
vote
0answers
62 views
uniformly convergent
Define functions $f_n\colon [0,1]\to\Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$ where $p$, $q>0$ and
$f_n\to 0$ pointwise on $[0,1]$ as $n\to \infty$ and $\sup|f_n(x)|=(n^{p-q})/e$.
Assume that ...
2
votes
1answer
62 views
How can I prove that $3$ is the only solution to the equation $2^n - 2n - 2 =0$ for $n\geq2$?
I'm working on a probability question:
Given the equiprobability of "having a boy" and "having a girl" as $1/2$ each, for what value of $n$ births, $n\geq2$ are the following two events independent?
...
3
votes
1answer
99 views
Hypergeometric functions inequality
Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers.
From a simple plot it looks like
$_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} ...
5
votes
0answers
223 views
expansion for $1-|t|$
Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions
$$
f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right)
$$
I am wondering if one can expand ...
2
votes
0answers
74 views
Iterated Root Mean Square-Arithmetic Mean
Can I find iterated Root Mean Square-Arithmetic Mean as a function of Arithmetic-geometric mean (AGM) with some transformations if it is possible?
if not possible, what is the closed form of it as ...
0
votes
0answers
311 views
Finding the Inverse of a Summation
I have seen more specific versions of this question but my question is more general. For any given summation does there exist an inverse. If not, how does one tell if the function has an inverse. ...
3
votes
2answers
119 views
Name of function: $\sum_{i=0}^N(N-i)$
I'm sure this is a really simple question but I have a series and I'm sure it has a name but my math is rusty and I'm not sure what it is.
$$ \sum_{i=0}^{N} (N - i)$$
I have an algorithm which has ...
1
vote
1answer
38 views
Formula to scale a series that is being bent with a root / power.
I have a reference number, Rx, and a series of numbers, Sx[], to compare to it. Let's call the output Ox[]. I am using a simple square root to accelerate the apparent difference between the reference ...
2
votes
2answers
297 views
closed form solution to a sum of functions
I need to find a closed form expression for the following:
$$\sum_{n=0}^{\infty}\tfrac{a^n}{(n!)(c-bn)}e^{(c-bn)t} \text{ with } a,b,c<1$$
By closed form expression, I mean a formula that can ...
1
vote
1answer
113 views
Is Bessel function $J_0(n)$ absolutely summable?
Is the Bessel function $J_0(n)$ absolutely summable i.e $\sum_{n=0}^{\infty}|J_0(n)| < \rm C$? Since $\lim\limits_{n \to\infty} J_0(n) = 0$, I'd assume the absolute sum converges to a constant ...
0
votes
1answer
158 views
what is the largest domain of a function on a real line?
hey guys i have function which is ln(x)
the domain that i know is when X >0;
the question is State the domain of f. Make it the largest possible domain on the real line.
does that mean like the ...
0
votes
1answer
332 views
If $\operatorname{Re}^{2}(x)=-1$, what is $x$?
$i=\sqrt{-1}$
$\operatorname{Re}(z)+i\cdot\operatorname{Im}(z)=z$
If $\operatorname{Re}^{2}(x)=-1$, what is $x$?
$x$ cannot be defined in complex number as $(a+ib)$. { $a$ and $b$ are real numbers ...
8
votes
1answer
170 views
Fourier Series involving the Jacobi Symbol
We know that the Fourier Series $$s(x)=\sum_{k\neq0}\frac{1}{k}\exp\left(2\pi ik x\right)$$ corresponds to the sawtooth function, $s(x)=\left\{x\right\} -\frac{1}{2}$. Suppose that ...
1
vote
2answers
98 views
Writing combined equation for alternating sequence
Suppose you have a sequence such as
$$5,1,11,5,17,9,23,13,29,17,35 ...$$
where a piece wise function can describe the sequence
$5+6(x-1)$ for x is odd, and $1+4(x/2-1)$ for x is even
Is there any ...
1
vote
5answers
282 views
Please help me to understand why $\lim_{n\to\infty } {nx(1-x^2)^n}=0$
This is the exercise:
"$f_{n}(x) = nx(1-x^2)^n, n \in {N}, f_{n}:[0,1] \to {R}. $ Find $ {f(x)=\lim\limits_{n\to\infty } {nx(1-x^2)^n}}$."
I know that $\forall x\in (0,1]$ $\Rightarrow (1-x^2) ...
3
votes
1answer
231 views
Laurent series for an infinite sum of functions
I have an infinite sum of analytic functions that is guaranteed to converge for every $x$, except for $x=0$:
\begin{equation}
g(x) = \sum_{n=1}^\infty f_n (x)
\end{equation}
I want to expand the ...
4
votes
2answers
1k views
Change of order of double limit of function sequence
The more general quesion is under what conditions the folloing equality will hold (all functions are real valued):
$$\lim_{x \rightarrow a} \ \lim_{j \rightarrow \infty} f_j(x) = \lim_{j \rightarrow ...
0
votes
2answers
154 views
How can I create an equation for a Gaussian distribution based on a sum of a series?
I am trying to create an equation that will generate a Gaussian distribution such that y = the sum of f(x) in a series of integers 1->28. I have y and want to know what the value of x at each integer ...
