Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).
0
votes
3answers
11 views
Which one is the correct series expansion?
Is $\quad p^{n+1}$ = $p^0+p^1+... + p^{n}$
or
$p^{n+1}$ = $p^0\times p^1\times... \times p^{n}$ ?
I am confused.
please explain the correct one.
0
votes
0answers
40 views
Radius of convergence and complex power series
I recall from calculus that the radius of convergence of a power series is the same for the derivative and the integral of that power series.
For complex, however, is this the same for derivatives ...
1
vote
1answer
83 views
Am I right or is Wolfram right?
Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies
$$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$
Determine a recurrence relation that ${a_n}$ satisfies.
I ...
1
vote
1answer
50 views
Writing a sum as a fraction
Express
$$\sum^{20}_{i=2}f(x)^i$$where $$f(x)=\sum_{i\geq 1}2^{i-1}x^{3i}$$ as a fraction of polynomials $p(x)/q(x)$ and simplify as much as possible.
Hmm. How to do it? Wolfram is really stupid on ...
0
votes
0answers
29 views
Solving $m$ in $m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}}$ from $n$ and $x$
How should one proceed in order to solve $m$, where $x$ is an integer
$$m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}} $$
from $n$ and $x$ in an unconditional form, such as, for ...
2
votes
1answer
37 views
Total area of squares.
We have a square whose length is $1$ unit. Every time we rotate by $\theta$ and scale the square such as you see in the image. Does the total area of squares converge if $\theta $ goes to $0$?
2
votes
4answers
77 views
How to use “results from partial fractions”?
Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies
$$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$
The denominator can be factored into $(1-2x)(1+x)^2$. Using ...
1
vote
2answers
42 views
Using mathematical induction to prove an identity related to combinatorics
Using Mathematical induction on $k$, prove that for any integer $k\geq 1$,
$$(1-x)^{-k}=\sum_{n\geq 0}\binom{n+k-1}{k-1}x^n$$
How should I proceed? The tutorial teacher attempted this question and ...
2
votes
1answer
37 views
Combinatorics of the Zeta function of a variety
I want to know if there is a good combinatorial interpretation of what the Zeta function of a variety $X$ over a finite field $\mathbb{F}_p$ counts. It is defined as $$\exp\sum N_j/j\,t^j,$$ where ...
1
vote
1answer
44 views
Is $e^z\sum_{k=0}^\infty\frac{k^3}{3^k}z^k$ analytic inside $|z|=3$?
Am I correct that the following function is analytic at least inside $|z|=3$? (I used the ratio test.) The solutions manual says that the function is analytic on and inside |z|=1, so I wonder if I'm ...
0
votes
1answer
35 views
power series quotient of polynomial functions
I have given $g(x)=\sum_{k=1}^\infty k^2x^k$. Why can you now write $g:(-1,1)\rightarrow\mathbb R$ as a quotient of two polynomial functions?
I just know the radius of convergence is ...
1
vote
4answers
43 views
Proving an identity using formal power series
4.
(a) Prove that $\dfrac{1-x^2}{1+x^3}=\dfrac{1}{1+\frac{x^2}{1-x}}$.
(b) By expanding each side of the identity in (a) as a power series, and considering the coefficient of $x^N$, prove ...
-1
votes
1answer
35 views
Express the following power series as a raional function
Consider the following power series:
$f(x) = \sum\limits_{i>=1} 2^{i-1}x^{3i}$ = $\ x^3 + 2x^6 + 4x^9 + ...$
$g(x) = \sum\limits_{i=2}^{20} f(x)^{i}$
Express both f(x) and g(x) as rational ...
1
vote
3answers
40 views
Laurent expansion problem
Expand the function $$f(z)=\frac{z^2 -2z +5}{(z-2)(z^2+1)} $$ on the ring $$ 1 < |z| < 2 $$
I used partial fractions to get the following $$f(z)=\frac{1}{(z-2)} +\frac{-2}{(z^2+1)} $$
then
...
0
votes
1answer
33 views
Find the first 5 terms of the expansion in a power series
Find the first 5 terms of the expansion in a power series
$$y′=xe^{x}+2y^{2}$$
I've got a riccati equation $$ x e^{x}+2y^{2}, y(0)=0$$
After solving: $$y=e^{x}(x-1)+\frac{2}{3}y^{3} - 1$$
And I ...
0
votes
1answer
43 views
Calculating a coefficient for a formal power series
My textbook has a whole bunch of exercises on finding some coefficient inside a formal power series. Unfortunately, there aren't any examples on how to do so, especially since many of the series ...
5
votes
1answer
28 views
Kernel of the evaluation map on a power series ring
Let $R$ be a commutative ring with unity and $r \in R$ a nilpotent element. Is it true that if $f \in R[[\epsilon]]$ satisfies $f(r) = 0$, then $(\epsilon - r) | f$ in $R[[\epsilon]]$? I tried solving ...
0
votes
2answers
20 views
Develop the next function:$f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0)$
Develop the next function:$\displaystyle f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0).$
For the first part:
$\displaystyle\frac ...
2
votes
1answer
43 views
Power series of $f(x)=\sqrt{\frac{1+x}{1-x}}$
How do I find the power series form of $\,f(x)\,$:
$$\displaystyle f(x)=\sqrt{\frac{1+x}{1-x}}$$
I tried to multiply the fraction by $\,\dfrac{1+x}{1+x}\,$ but it didn't help...
1
vote
2answers
59 views
Identity with Bernoulli numbers: $\sum\limits_{k=1}^{n}k^p=\frac{1}{p+1}\sum\limits_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j}$
How I can prove that
$$\sum_{k=1}^{n}k^p=\frac{1}{p+1}\sum_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j},$$
where $B_j$ is the $j$th Bernoulli number?
I hope to find the answer. Thanks for help.
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
$$
\sum_{n=0}^\infty ...
1
vote
3answers
44 views
Power series of $\frac{\sqrt{1-\cos x}}{\sin x}$
When I'm trying to find the limit of $\frac{\sqrt{1-\cos x}}{\sin x}$ when x approaches 0, using power series with "epsilon function" notation, it goes :
$\dfrac{\sqrt{1-cos x}}{sin x} = ...
5
votes
4answers
137 views
Are Taylor series and power series the same “thing”?
I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
5
votes
3answers
52 views
Compositions of $n$ with largest part at most $m$
This is a problem from Stanley's Enumerative Combinatorics that I'm failing at a bit (lot):
Let $\bar{c}(m,n)$ denote the number of compositions of $n$ with largest part at most $m$. Show that ...
0
votes
1answer
30 views
Radius of convergence - ratio test for power series/real numbers
Having trouble with when to apply the ratio test for power series and when to apply the ratio test for real numbers.
For example, find radius of convergence of these....
$\sum_{n=0}^{\infty}(-1)^n ...
6
votes
1answer
75 views
Existence of a power series converging non-uniformly to a continuous function
I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that:
$f$ converges and is continuous on the closed unit disk $D$ and
the series $\sum_n a_n z^n$ does not converge ...
2
votes
1answer
36 views
Hermite's equation of order $\alpha$
Show that the general solution of Hermite's equation of order $\alpha$:
$${y}''-2x{y}'+2\alpha y=0$$
$$is$$
$$y(x)=c_{0}y_{1}(x)+c_{1}y_{2}(x)$$
where $y_{1}(x)$ and $y_{2}(x)$ are power series ...
1
vote
2answers
41 views
Finding the $x^n$ coefficient of the power series $\sum\limits_{n=0}^\infty\frac{x^{2n+3}}{n!}$
I have a practice test question that asks:
Given the following Maclaurin series representation, $$\sum\limits_{n=0}^\infty\frac{x^{2n+3}}{n!}$$ what is the coefficient of $x^n$?
I have the ...
-1
votes
0answers
64 views
Treating indices as if they were exponents [closed]
Suppose $G(x) = a_0 x^0 + a_1 x^1 + a_2 x^2 + a_3 x^3 + \cdots$ and I wish to write
$$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$$
How can I ...
2
votes
1answer
22 views
Residue of a 1-form in a Riemann Surface does not depend of the chart
Let's suppose that $X$ is a Riemann Surface, $\omega$ is a meromorphic 1-form in $X$ and let $p$ be a pole of $\omega$ of order $M$. I want to show that the residue of $\omega$ at $p$, defined by
$$
...
1
vote
1answer
35 views
Determining power series for $\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$
I'm looking for the power series for $f(x)=\frac{3x^{2}-4x+9}{(x-1)^2(x+3)}$
My approach: the given function is a combination of two problems. first i made some transformations, so the function looks ...
1
vote
1answer
34 views
What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.
I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
0
votes
1answer
36 views
Expansion of power series for $\frac{\ln(1-x)}{1+x}$
My Problem is to expand $f(x)=\dfrac{\ln(1-x)}{1+x}$ into a power series.
My Approach: from looking onto the Graphs of this function, i know, for rising x the y is falling towards Zero, without ...
2
votes
3answers
48 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 $ ...
3
votes
1answer
89 views
Is $\pi$ to do with circles or power series?
To get straight to the point: is $\pi$ defined as the ratio of the circumference and diameter of a circle, or as the first non-zero root of the power series of $\sin{x}$?
If the former, then $\pi$ ...
2
votes
2answers
45 views
Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite
There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$
R := \left(\limsup_{n\to\infty} ...
3
votes
1answer
77 views
Can't solve this series…
I need to solve for the closed form of the following series:
$$
S_k(x)=\sum_{n=1}^{\infty} \frac {n} {n^2-k^2}x^n
$$
I can't seem to get it in terms of any known series. Differentiating, ...
2
votes
1answer
38 views
Radius of convergence of a power series with Bernoulli numbers
Say, we use the definition:
Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
and then derive power series representations of the ...
7
votes
5answers
241 views
When are we (not) allowed to replace $x$ by $ix$?
It seems to be quite a common manipulation to replace $x$ by $ix$.
Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
3
votes
1answer
39 views
Determining the Radius of Convergence for an alternating power series
I have a given alternating power series: $\sum\limits_{n=2}^{\infty}(-1)^{n}\frac{1}{n2^n}\cdot x^n$ and i need to find the Radius of Convergence.
I tried myself with this task and i found an answer, ...
0
votes
1answer
35 views
Solving the second order differential equation: y' =( x^2)*y as a power series
What is the proper way to do this problem as a power series? The way I'm doing it, I end up with a very complicated term.
how I'm doing it:
take the series for y and assume it's of the form ...
2
votes
2answers
58 views
How does one get the Bernoulli numbers via the generating function?
Here is the definition:
Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
I've tried to naively expand $\frac{x}{e^x-1}$ around ...
1
vote
1answer
43 views
Homework: Maclaurin Power Series Help
I'm trying to find the Maclaurin Power Series for
$$f(x)=\frac{3x-8}{3x^2+5x-2}$$
but each degree of differentiation gets more complex with no discernible pattern. Any help is appreciated, thanks.
1
vote
2answers
67 views
Expanding an analytic function to a powerseries
How would you expand the analytic function $$\frac{1}{1-z-z^2}$$ to a series of the form $$\sum_{k=0}^\infty a_k z^k \, \, ?$$
3
votes
1answer
53 views
A numeral system built around Dirichlet series, by analogy of how positional numeral systems are built around power series?
For any natural number and chosen base p, the number admits a unique expression of the form $a_np^n + ... + a_2p^2 + a_1p^1 + a_0$, where $a_k < p$ for all k. This property is effectively what ...
2
votes
2answers
37 views
Sine and Cosine Power Series
I have read that sine and cosine can be represented as power series. Power series, as I understand them, are infinite series that can be represented as:
$\sum_{j=0}^{\infty} a_j (x-x_0)^j$
where ...
1
vote
1answer
33 views
Points around which one expands and the radiuses of convergence
I'm trying to make sense of the following passage:
Let $f(x)=\frac{1}{x+1}$ and $R_0$ the radius of convergence of the Taylor series of $f$ around $x_0=0$, analogously: $R_1$ — around ...
2
votes
4answers
80 views
Radius of Convergence of power series of Complex Analysis
I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
3
votes
1answer
66 views
Continued fraction expansion related to exponential generating function
A recent SciComp.SE Question motivates us to ask for a nice continued fraction expansion of the following Maclaurin series:
$$ f(x) = \sum_{n=0}^\infty \frac{B_n\; x^{n+3}}{n! (n+3)} = \int_0^x ...
1
vote
2answers
52 views
find a solution from power series for multiple variable
$3^x4^y = 4,782,969 $ where $x$ and $y$ are integers. What is the value of $y$?
Is there any theory to solve this type problem?
i have tried to make $4,782,969$ into power series but couldn't. So a ...
