Questions tagged [power-series]
Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.
8,391
questions
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14
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Monotonicity of quotient of power series
How can it be proved that
$$
f(x) = \frac{\sum_{n=0}^{\infty} \frac{n}{2 x} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}}{\sum_{n=0}^{\infty} \frac{(\sqrt{x})^{n}}{\sqrt{n!}}}
$$
is monotonic?
The plot of $f(x)$ ...
0
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0
answers
14
views
Convergence of taylor series expansion in the evaluation point
The Taylor series expansion of $f(x)=e^x$ in $a=1$ is:
$T(x,a=1)=\sum_{n=0}^{\infty}\frac{e\left(x-1\right)^{\ n}}{n!}$
The interval of convergence using the absolute ratio test is $x\in(-\infty,\...
- 163
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votes
0
answers
35
views
Stanley 's example 1.1.11 and the exponential function
I opened Stanley's Enumerative combinatorics Vol. 1 and looked at example 1.1.11:
Suppose $F(0)=1$ and let $G(x)$ be the unique power series satisfying
$G'(x)=F'(x)/F(x)$, $G(0)=0$.
The author wants ...
- 245
0
votes
1
answer
50
views
Differentiating a Bessel function power series
For a function defined as
$$\begin{align}
y(x)
&:= \sqrt{\frac{\pi x}{2}} J_{1/2}(x) \\
&= \sqrt{\frac{\pi x}{2}} \sum_{k=0}^{\infty} \frac{(-1)^k}{k! \Gamma (k + 3/2)} \left( \frac{x}{2} \...
- 428
2
votes
1
answer
73
views
Absolute convergence on boundary implies continuity of power series
Let $f(z) = \sum_{n=0}^{+\infty}c_nz^n$ be a complex power series with radius of convergence $R=1$. Suppose that the series of coefficients converges absolutely, i.e. $\sum_{n=0}^{+\infty}|c_n| < +\...
- 1,125
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0
answers
52
views
Is there a closed form for the power series $\sum_{k\geq 1}x^{2^{k}}$? [closed]
I would like to know if anyone could suggest me a reference where the authors discuss and provide a closed form for the power series:
\begin{align*}
f(x) := \sum_{k\geq 1}x^{2^{k}}
\end{align*}
where $...
- 439
5
votes
0
answers
91
views
Summing a nonstandard sequence, closed form of $S_n(x) = \sum_{i=1}^n x^{c^{i-1}}$
Arithmetic sequences have a common difference, where you add a constant to each term to get the next. Geometric sequences have a common ratio, where you multiply a constant to each term to get the ...
0
votes
1
answer
81
views
Proof that $e^z=e^xe^{iy}$ [duplicate]
The proof goes like this:
$$
\begin{aligned}
e^z=e^{x+iy}&=\sum_{n=0}^\infty\dfrac{(x+iy)^n}{n!}\\
&=\sum_{n=0}^\infty\sum_{k=0}^n\dfrac{x^k(iy)^{n-k}}{k!(n-k)!}\\
&=\sum_{n=0}^\infty\...
2
votes
1
answer
78
views
Generalization of binomial coefficients to both non-integer arguments
It is known that binomial coefficients can be generalized to the following: for $s\in\mathbb R$ and $k\in\mathbb N$,
\begin{equation*}
\binom{s}{k} := \prod_{i=0}^{k-1} \frac{s-i}{k-i} = \frac{s(s-1)...
- 1,932
1
vote
0
answers
56
views
Behavior of Incomplete Gamma function with negative s, as x goes to zero
From the second item under the Asymptotic Behavior section of the Wikipedia article for the incomplete gamma function, it is written that
$$
\frac{\Gamma(s,x)}{x^s} \to -\frac{1}{s} \text{ as } x \to ...
- 271
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votes
1
answer
56
views
Calculate the powers of the sets.
Find the powers of the sets:
(a) 𝑋 = {$𝐴 : 𝐴 ⊆ \mathbb R$ and $𝐴$ has a smallest and largest element};
(b) 𝑌 = {$𝐴 : 𝐴 ⊆ \mathbb Z$ and $𝐴$ has a smallest and largest element};
(c) 𝑍 = {$𝐴 : ...
- 25
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0
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44
views
Can a nonzero complex power series have an uncountable set of complex roots?
Following Can a real power series have an uncountable number of real roots? and this essentially equivalent question about a sort of linear independence of powers of a real function, the natural thing ...
- 504
1
vote
2
answers
53
views
In what sense, Taylor series at $x=0$ is better than Taylor series at $x=a$, and vice-versa?
In the book `Calculus Early Transcendentals (6th Edition)' by James Stewart, on p. 739, the author writes:
We have two series representations for $e^x$ as follows:
$$e^x=\sum_{n=0}^\infty x^n/n! , \...
- 11
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votes
2
answers
82
views
Why does this sum not diverge?
So I'm asking this question in general but with a motivating example:
Say we have a function $$f(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$
So both the top ($x^n$) and bottom ($n!$) of this function grow -...
1
vote
2
answers
85
views
Computing integral of a series [closed]
Can anyone help solve this problem or provide any advice please?
Compute the integral of $f: [0,1] \rightarrow \mathbb{R}$ defined by:
$$f(x) = \begin{cases}
0 & \text{if }\; x = 0, \\
...
- 19
4
votes
2
answers
143
views
Seeking more alternate proofs of a combinatorial generating function identity $G(x)=\overline{G}(-x)^{-1}$ related to counting strings.
Let $\mathcal{S}=[m]^*$ be the set of all strings on the alphabet $[m]=\{1, 2,\cdots, m\}$. Let $\Sigma\subset[m]^2$ be a set of strings of length $2$, and let $\overline{\Sigma}=[m]^2\backslash\Sigma$...
- 2,892
1
vote
1
answer
113
views
Can $I[[X]] = I\cdot R[[X]]$ if $I$ is not finitely generated?
Let $R$ be a ring and $R[[X]]$ be the formal power series ring. For any ideal $I\subset R$, denote by $I[[X]]$ the ideal of power series with coefficients in $I$.
If $I$ is finitely generated, it is ...
- 766
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0
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97
views
Why is $\ln2\approx 0.4^{0.4}$?
Just stumbled upon
$$\ln2\approx 0.4^{0.4}$$
and wondered if that's just a coincidence, or whether there's some deeper reason?
$$\ln2 - 0.4^{0.4}\approx 0.00000234$$
which is a relative error of just $...
- 9,985
2
votes
1
answer
41
views
Solving ODE using power series method but get the general solution $y=0$
Given ODE
$$x^2y'-y=0,\qquad x\neq 0.$$
Solve given ODE using power series method.
Let $$y=a_0+a_1x+a_2x^2+a_3x^3+\ldots.$$
Then,
$$y'=a_1+2a_2x+3a_3x^2+4a_4x^4+\ldots.$$
Now, we substitute to ODE ...
- 2,296
1
vote
1
answer
53
views
Prove that a functional series satisfies a particular functional equation
I have some difficulties with this problem. Let
$$g(x)= \lim_{k\to\infty}\sum_{n=-k}^k \frac{1}{x+n}$$
Prove that
$$g\left(\frac{x}{2}\right)+g\left(\frac{x+1}{2}\right)=2g(x)$$
I've already proved ...
4
votes
1
answer
115
views
How to define $e$ from power series?
I'm trying to take a different approach to show the exponential function and $e$ should exist in calculus. Say, we want there to be a hypothetical function $f$ which should have the property that it ...
- 189
1
vote
1
answer
87
views
Calculating the sum of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$ [duplicate]
I want to find the sum of $\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$.
I have tried to turn it into a power series for a known function, with no luck. I also tried to write it as $\sum_{n=0}^{\infty} \...
- 13
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0
answers
28
views
why does the formula for the radius of convergence $R^{-1}=\lim_{n\to \infty}\frac{|a_{n+1}|}{|a_n|}$ valid here?
Suppose we have the series of the form $\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}$. How do I find the radius of convergence?
My attempt
I used the Ratio test. I got the R=1.
But in the textbook ...
- 593
1
vote
0
answers
51
views
A power series solution to wave equation
Find a solution of
$$ u_{tt}-u_{xx}=\lambda^2u$$
of a form
$$u=f(x^2-t^2)=f(s) \text{ with } f(0)=1,$$ in form of power series in s, where $\lambda$ is a nonzero constant.
First thing I do is to find $...
- 79
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0
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56
views
Domain of convergence for a power series
I have a homework question that we have not discussed at all how to do in class, so any help or explanations would be greatly appreciated! I don't know where to start.
In each of the following cases, ...
- 1
0
votes
1
answer
23
views
Series bounded by uniform convergent series is uniform convergent?
I was reading about the Weierstrass $M$-test, and I was wondering about the following:
Suppose that $(f_{n})$ is a sequence of real-valued functions defined on a set $A$, and there exists $(g_{n})_{n ...
- 1,280
2
votes
2
answers
93
views
Prove that tan($Z$) = $\frac{e_1t - e_3t^3 + e_5t^5 - \cdots}{1 - e_2t^2 + e_4t^4 - \cdots}$
Let $e_r$ and $p_r$ denote the $r$-th elementary symmetric
function and power sum, respectively. Let $t$ be a formal variable
and define
$$Z := p_1t-p_3t^3/3+ p_5t^5/5- \cdots $$
Prove that the ...
- 81
6
votes
2
answers
186
views
Series with only positive integer coefficients
I want to show that the coefficients of the series
$$
f(x) = \sqrt[4]{\frac{1+4x}{1-4x}} = 1 + 2x + 2x^2 + 12x^3 + \cdots
$$
are only positive integers. Since differentiating $f(x)$ shows $\left(1 - ...
- 63
3
votes
1
answer
98
views
I can't find the mistake in my proof
I thought about something in power series and I proved a theorem that I have never seen before:
The theorem:
Let $\begin{align} \sum_{n=1}^{\infty} a_nx^n \end{align}$ be a power series, with a ...
- 456
-1
votes
0
answers
18
views
Are there generalized power-reduction formulae? [duplicate]
I am studying trigonometry and have wondered if there are generalized power-reduction formulae for sin^n(x) and cos^n(x) yet the best that I found was formulae for odd and even powers on Wikipedia ...
- 1
2
votes
3
answers
142
views
proof for series to the power of 5 is divisible by n+1 for even n and n congruent to 3 mod 4
I really need help with this:
$$\sum_{x=1}^{n} x^5 $$ is divisible by $n+1$ when $n$ is even, and when $ n \equiv 3 \pmod{4} $.
What I first did was proving by induction that $$\sum_{x=1}^{n} x^5 = \...
- 23
2
votes
2
answers
99
views
Power series of $\frac{1}{7+x}$
For my calculus 2 class, I was given the problem to solve for the power series of $\frac{1}{7+x}$. For my solution, I said $\frac{1}{7+x} = -\frac{1}{1 - (8 + x)}$ and used $8 + x$ as $r$ and $-1$ as $...
- 21
5
votes
1
answer
106
views
Series expansion of the integral $\int_{-\infty}^{\infty} \frac{\tanh{(x^2-a^2)}}{x^2-a^2} dx$ in the limit of small $a$
I found that when $a \gg 1$, the function $I(a) = \int_{-\infty}^{\infty} \frac{\tanh{(x^2-a^2)}}{x^2-a^2} dx \approx \frac{2\gamma + 2 \log(16/\pi) + 4 \log a}{a}$.
How does the integral behave when $...
- 2,096
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votes
0
answers
96
views
Prove by induction: Let $r \in \mathbb{N}$. $\exists a_{r,1}, \ldots, a_{r,r} \in \mathbb{Q} \, \forall n \in \mathbb{N}$
$$
\sum_{k=1}^{n} k^r = \frac{1}{r+1} n^{r+1} + a_{r,r} n^r + \ldots + a_{r,1} n
$$
I'm apparently supposed to use the binomial theorem, but I can't seem to figure it out. The note says: "For the ...
- 1
0
votes
2
answers
106
views
Compute the sum of $\sum_{n=1}^\infty \frac{1}{(2n-1)3^n}$ [closed]
I was asked to sum up the series:
$$\sum_{n=1}^\infty \frac{1}{(2n-1)3^n}$$
However, I tried to write it：
$$\sum_{n=1}^\infty \frac{1}{(2n-1)3^n}=\frac{1}{1*3} + \frac{1}{3*9} + \frac{1}{5*27}+...$$
I ...
- 31
4
votes
0
answers
62
views
Apply Power Series Solution Method on Solving ODE with non-integer x order term $ \frac{d^2y}{dx^2}+x^{1/2}y=0 $
Problem:
Use series solution expansion to solve ODE in region x>0
$$ \frac{d^2y}{dx^2}+x^{1/2}y=0 $$
Attempts:
Assume $y=\sum^\infty_{n=0} a_nx^n$, then $\frac{d^2y}{dx^2}=\sum^\infty_{n=2} n(n-1)...
- 41
2
votes
1
answer
98
views
Is there such a thing as an intermediate value theorem in complex analysis?
I have an exercise I want to solve, but got stuck in the second part due to not having something like the intermediate value theorem for the complex plane. The exercise is: Let $\Omega$ be an open set ...
0
votes
3
answers
201
views
For holomorphic function $f: \Omega \to \mathbb{C}$ with $f^{(k)}(z_0) \in \mathbb{R}$ prove that $f(x) \in \mathbb{R}$ for every $(z_0 - r, z_0 +r)$
I have some difficulties with a question I have come across.
The question goes as follows:
Let $\Omega$ be an open set with $z_0 \in \Omega \cap \mathbb{R}$. Let $f: \Omega \to \mathbb{C}$ be ...
- 11
2
votes
0
answers
34
views
On the naturality of the definition of a deformation?
Let $(A,\mu_0)$ be an associative unital $k$-algebra over a field $k$. A formal deformation of $(A,\mu_0)$ is sometimes defined as a $k[[t]]$-bilinear map $\mu: A[[t]]\times A[[t]]\rightarrow A[[t]]$ ...
- 1,653
1
vote
1
answer
59
views
Operator inverse as a geometric series [duplicate]
Let us consider an operator in the form
$$
\left( \mathbf{1} - \lambda \hat{O} \right) \,.
$$
Under which circumstances I can write its inverse as
$$
\left( \mathbf{1} - \lambda \hat{O} \right)^{-1} = ...
- 13
0
votes
0
answers
60
views
Holomorphic extension of the power series $\sum_{n \geq 0} z^n $
So the power series $\sum\limits_{n \geq 0} z^n $ converges in the unit disc and is holomorphic in the unit disc as well with the derivative $\sum\limits_{n \geq 0} nz^{n-1}$. I am trying to prove ...
- 303
4
votes
1
answer
68
views
Example of $a_k$ such that $\sum a_k$ doesn't converge but $\lim_{x \rightarrow 1, x< 1} \sum a_k x^k$ exists.
In class we were taught that if $\sum a_k$ converges, then $\lim_{x \rightarrow 1, x < 1} \sum a_k x^k$ exists. The proof is based on the Cauchy's convergence criterion:
if the sum converges, then $...
- 239
1
vote
0
answers
58
views
How to calculate convergence of a series related to the Euler's constant?
I understand that
$$
\sum_{r=1}^n \frac{1}{r} \approx \int_{1}^n \frac{dx}{x} = \log n
$$
I would like to verify whether the following holds true as well, for a series that is related to the above ...
- 233
1
vote
1
answer
120
views
Power-series expansion of $\mathrm{e}^{\frac{z}{2}(t-\frac{1}{t})}$
I would like to know the procedure to do the power-series expansion of this function:
$$\LARGE{\mathrm{e}^{\frac{z}{2}(t-\frac{1}{t})}}$$
so that I would get this:
$$\sum_{r=0}^\infty \sum_{s=0}^\...
- 3,872
1
vote
0
answers
53
views
How to convert first order singular ODE to integral equation?
I have the following ODE:
\begin{equation}
v(x)y'=a(x)y^2+b(x)y,~~~y(0)=\varphi,~~v(0)=v(1)=0.
\label{eq1}
\end{equation}
I know that for a similar equation of the form $\varepsilon y'=a(x)y^2+b(x)y~~$...
1
vote
0
answers
70
views
Show that $k[[x,y]]/(y^2-x^2) \simeq k[[x,y]]/(y^2-x^3-x^2)$
I have to show that $$k[[x,y]]/(y^2-x^2) \simeq k[[x,y]]/(y^2-x^3-x^2)$$ where $k[[x,y]]$ is the ring of formal series in $x,y$ over an algebraically closed field $k$.
I have the hint that the map $y \...
- 4,500
0
votes
0
answers
75
views
$k[[t]]$-module structure on coordinate ring
Let $k$ be a field and $k[[t]]$ the ring of formal power series over $k$. A text that I am reading says:
Consider the coordinate ring $k[x,y,t]/(x^2+txy)$ with its canonical $k[[t]]$-module structure....
- 1,653
0
votes
0
answers
41
views
Expressing $ \sum_{\lambda\in\text{Spec}(\mathbf{X})} \lambda \log \lambda$
Say $\text{Spec}(\mathbf{X})$ is a set of eigenvalues of a real positive definite matrix $\mathbf{X}$.
How do I express
$$ \sum_{\lambda\in\text{Spec}(\mathbf{X})} \lambda \log \lambda$$
without ...
- 525
0
votes
0
answers
63
views
Series of Analytic Functions is Analytic
Let $0 \in \mathbf{N}$. Let $P_m(x): [0,1] \to \mathbf{C}$ be bounded analytic functions for every $m\in \mathbf{N}$. Formally, define
$$
f(x) = \sum_{m\in \mathbf{N}}c_m P_m(x)\overline{P_m}(x),
$$
...
- 65
-4
votes
1
answer
163
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Solution of $ 1 + \frac{x}{2!} + \frac{x^2}{4!} + \frac{x^3}{6!} + \frac{x^4}{8!} + \dots = 0$ [closed]
So according to the book, the solution is true but is this even possible?
The equation is:
$$1 + \frac{x}{2!} + \frac{x^2}{4!} + \frac{x^3}{6!} + \frac{x^4}{8!} + \dots = 0 \ $$
It asks the solutions, ...
- 13