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).
4
votes
2answers
200 views
Proving the equivalence of a sum and a double integral
Based on "Certain Subclass of Starlike Functions" journal by Chun-Yi and Shi-Qiong Zhou in 2007 (Science Direct), I found difficulties to understand the proof in Theorem 3 where they have verified:
...
4
votes
3answers
295 views
A deceiving Taylor series
When we try to expand
$$
\begin{align}
f:&\mathbb R \to \mathbb R\\
&x \mapsto
\begin{cases}
\mathrm e^{-\large\frac 1{x^2}} &\Leftarrow x\neq 0\\
0 &\Leftarrow x=0
...
3
votes
3answers
175 views
the sum of a series
I am stuck on the computation of the following sum:
$$\sum_{k=0}^{\infty} {\Big( {\frac{q}{k+1}} \Big)}^k ,$$
where $k$ is a natural number, and $0<q<1$.
3
votes
3answers
241 views
Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$
Given the geometric series:
$1 + x^2 + x^4 + x^6 + x^8 + \cdots$
We can recast it as:
$S = 1 + x^2 \, (1 + x^2 + x^4 + x^6 + x^8 + \cdots)$, where $S = 1 + x^2 + x^4 + x^6 + x^8 + \cdots$.
This ...
3
votes
2answers
313 views
How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$
I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
3
votes
1answer
273 views
Formal Power Series — what's in it?
I have the following statement in a paper:
Let $\Psi$ be the formal power series defined over the alphabet $\Omega$ and the log semiring by: $(\Psi, (a, b)) = -log(c((a,b)))$ for $(a,b) \in \Omega$, ...
2
votes
2answers
134 views
Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?
I'm looking for an intuitive understanding instead of a formal proof. Thanks for the help.
2
votes
1answer
161 views
Riemann Zeta Function Manipulation
The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$
(i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty ...
2
votes
2answers
439 views
Help on differential equation $y''-2\sin y'+3y=\cos x$
$y''-2\sin y'+3y=\cos x$
I'm trying to solve it by power series, but I just can't find the way to get $\sin y'$. Is there any special way to find it?
1
vote
1answer
214 views
The coefficients in the inverse of unit lower triangular Toeplitz matrix
I have a question about the inverse of lower triangular Toeplitz matrix.
There is a matrix $Q$ which can be infinite-dimensional.
$$
Q=\left[\begin{array}{ccccc}
1\\
-k_{1} & 1\\
-k_{2} & ...
0
votes
1answer
128 views
What's the limit of coefficient ratio for a reciprocating power series?
I have a question about the coefficient in the inverse of the power series.
Assume
$$
f=1-\sum_{i=1}^{\infty}(ck_i)x^i,
$$
where $c$ and $k_i$ are positive and $0<ck_i<1$ for any $i>0$. ...
0
votes
2answers
86 views
Trying to revert a series with problematic log term
I'm stuck on a problem which I'm not sure has a solution. I have the first few terms of a series I want to invert,
$y(x)=\ln(x)+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$
I know the inverse exists ...
4
votes
2answers
136 views
What's the background of this exercise?
I found this interesting exercise on a calculus book (Stewart)
Let
$$
u=1+\frac{x^3}{3!}+\frac{x^6}{6!}+\cdots
$$
$$
v=x+\frac{x^4}{4!}+\frac{x^7}{7!}+\cdots
$$
$$
...
3
votes
4answers
193 views
Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.
Show that $\displaystyle\lim_{n\to\infty} nx^n=0$ for $x\in[0,1)$.
3
votes
2answers
247 views
Why is Taylor series expansion for $1/(1-x)$ valid only for $x \in (-1, 1)$?
After finding an expansion of $$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$
a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$.
...
3
votes
2answers
144 views
Power Series $0^{0}$
My textbook explains that the power series: $\sum_{n=0}^{\infty} x^{n}/n!$ converges for $x=0$ because the terms of the series get the value 0.
My problem with this argument is the first term, ...
3
votes
4answers
284 views
Question Regarding The Power Series For $e^x$
Currently I'm reading Higher Engineering Math by John Bird and under exponential function he talks about obtaining the value of $e$.
He begins by saying
The value of $e^x$ can be calculated to ...
3
votes
6answers
284 views
how to find this generating function
this is the power series:
$$\sum_{i=0}^\infty n(n-1)^2 (n-2) z^n.$$
how can I find a generating function from it? I could use the third derivative but the $n-1$ is squared so I don't know what to ...
3
votes
5answers
299 views
Calculate sum of an infinite series
I have been struggling with this functional series. $$\sum_{n=1}^{\infty}{(-1)^{n-1}n^2x^n}$$ I need to calulate the sum.Any tips would be appreciated.
3
votes
2answers
724 views
Formal Proof of Exponential rule
I tried to prove this, but was unsuccessful for a long time..
Any ideas?
Prove that $(\exp(x))^y=\exp(xy)$ using the identities,
$$\exp(x)=\sum_{n\geq0} \frac{x^n}{n!}, \quad ...
2
votes
2answers
61 views
An approximate solution to an ODE
I am interested in the ODE:
$x^\prime = x^2 + t^2$
$x(0)=0$
The power-series method is not (easily?) applicable here. Do you have any suggestions how to solve it?
2
votes
1answer
140 views
How to find the general solution of $(1+x^2)y''+2xy'-2y=0$. How to express by means of elementary functions?
Find the general solution of $$(1+x^2)y''+2xy'-2y=0$$
in terms of power series in $x$. Can you express this solution by means
of elementary functions?
I know that $y= ...
2
votes
1answer
471 views
Frobenius Method to solve $x(1 - x)y'' - 3xy' - y = 0$
So, Im trying to self-learn method of frobenius, and I would like to ask if someone can explain to me how can we solve the following DE about $ x = 0$ using this method.
$$ x(1 - x)y'' - 3xy' - y = 0 ...
2
votes
3answers
69 views
If $a_0\in R$ is a unit, then $\sum_{k=0}^{\infty}a_k x^k$ is a unit in $R[[x]]$
Let $R$ a ring, and let $$\displaystyle R[[x]]=\left\{\sum_{k=0}^{\infty}a_k x^k\;\middle\vert\; a_k\in R\right\}$$ with addition and multiplication as defined for polynomials. We have that $R[[x]]$ ...
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.
1
vote
2answers
288 views
Radius of convergence of a sum of power series
I have two series
$\displaystyle\sum_{n=1}^{\infty} a_n x^{n}$
$\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$
with radius of convergence $2$ and $3$ respectively.
How can I find the radius of ...
1
vote
0answers
71 views
series look up site
Is there a site for looking up a series to see some of the associated functions. (In the spirit of Encyclopedia of Integer Sequences OEIS.)
In particular I am looking for functions related to $ \sum ...
1
vote
3answers
106 views
Formula to $\ln$ that holds on interval $x \geq 1$
In the Wikipedia we can find two formulas using power series to $\ln(x)$,
but I would like a formula that holds on the interval $x \geq 1$ (or is possible to calculate $\ln(x)$ to $x \geq 1$ with the ...
0
votes
1answer
87 views
Binary relationship between powers and sum of powers.
I want to optimize a function that determines whether a given number $n$ is EITHER (a power of 2) OR (the sum of powers of 2). Using, this answer, it appears that a sum of power of 2s contain at most ...
0
votes
1answer
201 views
Power series expansion
I recently had a problem. I know how to evaluate power series but I cannot seem to find an expansion for $\sqrt{x+1}$.
I've tried differentiating it, in order to bring it in reciprocal form but that ...
0
votes
1answer
85 views
How can we take a power series and multiply each term, i.e. $c_n x^n$ by $y^n$?
In other words, given a power series $f(x)$, is there an alternative to taking $\lim_{x\to{x y}}f(x)$? I ask this because I thought that there may be a way to replace the limit by integration, or ...
0
votes
0answers
73 views
Is there any way to create (a closed form for) this power series/generating function?
There is a fairly simple pattern to it.
$$1 y + $$
$$(1 + 1x)y^2+ $$
$$(1+1x+1x^2 + 1x^3)y^3 + $$
$$(1+1x+\dots+1x^7)y^4 + $$
$$(1+1x+\dots+1x^{15})y^5 + $$
$$\dots$$
Does anyone know of a way ...
