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).

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2
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1answer
57 views

power series for matrix with elements smaller than 1

If I have a square matrix A such that all elements $|a_{ij}| < 1$ does this guarantee that all my eigenvalues will also be less than 1 and that the power series $S = I - A + A^2 - A^3...$ will ...
0
votes
3answers
56 views

Problem about ODE and power series

For each $a \in \mathbb{Z}^+$ let the following ODE $$ x'' - \dfrac{a (a+1)}{(1 +t^2)} x = 0$$ Using power series around the origin, show that the equation has a solution $p_a(t)$ which is a ...
1
vote
1answer
61 views

Solve ODE using analytic solutions

Let the following ODE: $x'' + tx' + x = 0.$ Find the general solution $x(t) = a_0 x_1(t) + a_1 x_2(t),$ with $a_0, a_1 \in \mathbb{R}$ and $x_1(t), x_2(t)$ are $t$ power series convergent for ...
1
vote
1answer
331 views

Analytic continuation of complex square root

This example comes from the book by Elias Wegert: Visual Complex Functions. Consider the function $f(z):=z^{1/2}$ for $z \in \mathbb{C}$ with $|z - 1| < 1.$ For these $z$, the function can be ...
1
vote
0answers
224 views

Find the indicial equation of $(x+2)^2(x-1)y''+5(x-1)y'-\pi(x+2)y = 0$

Find all singular points of each equation, and determine whether they are regular or irregular. At each regular singular point, find the indicial equation and the exponents of singularity. ...
2
votes
3answers
433 views

Find the form of a second linear independent solution when the two roots of indicial equation are different by a integer

Consider the differential equation $$x^2y''+3(x-x^2)y'-3y=0$$ $(a)$ Find the recurrence equation and first three nonzero terms of the series solution in powers of $$ corresponding to the larger root ...
28
votes
2answers
834 views

How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

Prove that there is no continuous functions $f:[0,1]\to R$, such that $$ f(x)+f(x^2)=x. $$ My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all ...
2
votes
1answer
205 views

Find one series solution for $xy'' - y = 0$

I have found the recurrence relation to be $a_{n+1} = \frac{(a_n)}{(n+1)(n)}$ . I am stuck at this part because no matter what I set the initial a to be, the following term will have a problem due to ...
0
votes
1answer
107 views

$\lim_{n\rightarrow +\infty} \frac{a_{n}}{a_{n+1}} = z_{0}$ with $z_{0}$ pole [duplicate]

This is an exercise from Stein-Shakarchi. Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole at $z_{0}$ on the unit circle. Show that if $f(z) = ...
0
votes
1answer
77 views

Determining radius of convergence $f(z)=\frac{\mathrm{e}^z}{z-1}$?

Can somebody help me with determining the radius of convergence of the power series of the following function $$f(z)=\frac{\mathrm{e}^z}{z-1}$$ about $z=0$?
0
votes
1answer
79 views

Integrating a Taylor series term-by-term

Why is $$\int_{0}^{z} \frac{\sin x}{x} \ dx =\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} \int_{0}^{z} x^{2n} \ dx$$ not valid for $z= \infty$? Well, at least I'm assuming it's not valid since ...
0
votes
2answers
41 views

Calculation of a power series sum

How can I calculate the following sum: $$\sum_{n=1}^\infty (n+2)x^n$$ What is wrong with spreading it to: $2x^n + nx^n$? both I know how to calculate. Thank you
1
vote
2answers
2k views

Sum of a power series $n x^n$ [duplicate]

I would like to know: How come that $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$ Why isn't it infinity?
0
votes
1answer
51 views

Power series (representation) of given function

Well I'm wondering if below power series is the correct result of the function - wolfram alpha doesn't give anything like the result. The function $$\frac{x}{2x^2+1} = x\cdot \frac{1}{1- \left ( ...
0
votes
1answer
47 views

Power series convergence question

Does there exist a sequence $c_{n}$ of complex numbers such that $$ \sum_{n=0}^{\infty} c_{n} z^{n} $$ has radius of convergence $R = \infty$, but for all other sequences $c_{n}'$ of complex numbers ...
0
votes
1answer
61 views

Series expansion of quotients

I'd like to start of with a simple formula from a textbook $T(W) = \frac{W^5}{1-2W} = W^5 + 2W^6 + 4 W^7 + \dots + 2^j W^{j+5} + \dots$ Obviously, this is an expansion of the quotient into a power ...
1
vote
1answer
39 views

Expanding One Function in Powers of Another

One sees here that it is possible to expand $f(x) = 2x^3 + 7x^2 + x - 6$ in powers of $x - 2$ by taylor expanding $f(x) = f(x - 2 + 2) = f(2 + h)$ about $2$, and this idea can be used in deriving the ...
5
votes
1answer
72 views

Is $\cos x$ irreducible as a power series?

Let $\mathbb{Q}_{\mathrm{ent}}[[x]]$ be the ring of entire functions with rational coefficients. Is $$ \cos x \;=\; \sum_{n=0}^\infty (-1)^n\!\frac{x^{2n}}{(2n)!} $$ irreducible in ...
2
votes
2answers
98 views

Rate of convergence of binomial series

This is the binomial series: $$(1+x)^\alpha = \sum_{k=0}^\infty \binom{\alpha}{k} x^k$$ where $|x|<1$ and $\alpha$ can be a complex number in general. How fast does it converge? I need an upper ...
2
votes
1answer
53 views

Clever way to expand 1/(z^2-n^2) in power series?

Is there a good trick to prove the following identity? $$\frac 1 {z^2-n^2} = -\sum_{i=0}^\infty \frac {z^{2i}} {n^{2(i+1)}}$$ I tried writing out the coefficients as a Taylor series, but this was ...
1
vote
2answers
168 views

$L^p$ spaces and counting measure

currently I am working on the following two exercises as a revision for my exam. Let $\mu$ be the counting measure on $\mathbb N$. Show that if $1 \le p < s < \infty$ then $f \in L^p$ implies ...
0
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2answers
95 views

Solving differential equation by using power series.

Find, using the power series: $$y(x)=\sum_{k=0}^\infty a_{k}x^k$$ a solution for the following differential equation: $$y'(x) = -x^2y(x),\,\, y(0)=1$$ What's the convergence radius of the constructed ...
2
votes
1answer
90 views

Why don't power series methods work for linear ODE's with singularities?

My math class tells me power series methods don't work for equations of the form $$f'' +p(x)f' +q(x)f = 0$$ if the functions $p(x)$ or $q(x)$ have singularities at the point about which you're ...
2
votes
2answers
74 views

How can i evaluate this power series?

$\sum_{n=0}^{\infty }\frac{1}{2n+1} \left (\frac{1}{3} \right )^{n}\left ( -1 \right )^{n} $ it's solved by power series of arctan. is it possible the answer written by real number?
0
votes
2answers
47 views

how can I sove approximation evaluation of this integral?

$$\int_{-1}^{0}\sin(e^{x})\,dx $$ approximation of this formula up to difference(error) $1/5000$ Because of the error size $1/5000$ , I think it's solved by taylor expansion.
2
votes
2answers
39 views

Computing the limit of function containing a power series.

Prove that if the sequence $a_{n}$ of real numbers converges to a finite limit; \begin{align} \lim_{n \rightarrow \infty} a_{n} = g, \end{align} then \begin{align} \lim_{x \to \infty} \left({\rm ...
0
votes
1answer
39 views

Ratius convergence of composition of series powers [closed]

For a complex function $f(z) = 1 / (2 + \exp(z))$, I have to find the radius of convergence centered in $1+i$ of the power series $f$
0
votes
2answers
98 views

$ \sum_{n\geq1}(\frac{1}{(n+1)!} ∏_{k=1}^{n} f(k))$ converge or diverge?

Let $f: \mathbb N - {0} \to \mathbb N -{0}$ injective function, verify is the serie $ \sum_{n\geq1}(\frac{1}{(n+1)!} ∏_{k=1}^{n} f(k))$ converge or diverge . I prove for $n=5$ and $n=9$ and found ...
0
votes
1answer
89 views

solve this differential equation using laplace transform and the series method :

Problem : $y''+8ty'-16y = 3 , y(0) = y'(0) = 0 $ I am supposed to use the series method to get F(s) , then get the inverse laplace transform to get f(t). I got the Laplace transform : $(s^3 - 24s) ...
4
votes
1answer
58 views

Optimal series expansion for “invertability”

Motivation: Often when dealing with physical phenomena, deviations from the model must be considered, so a variable, say $x\in[0,1]$ will be replaced by a power series expansion: $$x'\ \to \ x(1+k ...
0
votes
1answer
64 views

Evaluating a series with some given formula [duplicate]

I have a formula for the power series corresponding to the function $$\frac{z^{3k}}{(3k)!}$$ and I need to evaluate a new series with it but I can't see how to manipulate it even though I've had some ...
2
votes
1answer
95 views

Find the sum of $\sum_{n=1}^\infty \frac{x^{n-1}}{3^nn}$ - What is wrong with my solution?

I have to find the sum of the following power series: NOTE: please assume that x is in the convergence domain. $$\sum_{n=1}^\infty \frac{x^{n-1}}{3^nn}$$ My ...
5
votes
3answers
185 views

Series of inverses of binomial coefficients

Can you think of a simple way of proving that $$ \sum_{n=k+1}^\infty \frac{1}{n \choose k} $$ is rational for any $k \geq 2$? Here's the background. Consider a series: $$ \sum_{n=1}^\infty ...
-1
votes
2answers
64 views

Power series with radius of convergence 2 that diverges at both -2 and 2?

I'm looking for a real power series that has radius of convergence 2 but diverges at both 2 and -2. Any idea? Thank you!
3
votes
1answer
188 views

Finding a radius of convergence of power series

I have to find the radius of convergence of some power series but I find myself in trouble for three of them : the series are $\sum2^kx^{k!}$ $\sum\sinh(k)x^k$ $\sum\sin(k)x^k$. For the first ...
1
vote
2answers
66 views

Finding power series

I need to find the power series for $e^z + e^{az} + e^{a²z}$ where $a$ is the complex number $e^{2πi/3}$. I know that $1 + a + a² = 0$. I have tried to differentiate the expression and give values ...
1
vote
0answers
78 views

Alternative Bound on a Double Geometric Series

If $|a_{mn}x_0^my_0^n| \leq M$ then a double power series $f(x,y) = \sum a_{mn} x^m y^n$ can be 'bounded' by a dominant function of the form $\phi(x,y) = ...
1
vote
1answer
78 views

Ratio of coefficients for Laurent series expansions [duplicate]

Let $f$ be analytic in the disk $D(0,2)$ except for a pole of order $1$ at $z=1$, and let $$f(z)=\sum_{k=0}^\infty a_k z^k$$ be the series expansion for $f$ in the disk $D(0,1)$. Prove that ...
1
vote
2answers
57 views

convergence ratio of the serie $e^{xn}$

How can I determine the values of $x$ such that the series converge: $$\sum_{n=0}^\infty e^{xn}$$ I'm really lost in this problem, please help.
1
vote
2answers
1k views

Converge of the sum $\sum_{k=1}^{n} k x^k $

For what values ​​of x the sum converges and what is the limit when $n \rightarrow \infty$ $\sum_{k=1}^{n} k x^k $ My work: First i try to calculate the interval and radius of convergence of ...
2
votes
3answers
115 views

power series expansion of $z^a$ at $z = 1$

I'm working through some problems in a complex analysis book, and one asks to compute the power series expansion of $f(z) = z^a$ at $z = 1$, where $a$ is a positive real number. The series should ...
2
votes
2answers
111 views

Expansion and convergence of $\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}$

Consider the series: $$\sum_{m=1}^{\infty}\frac{\sin(2\pi n x^{1/m})}{ m}\;\;\;\;n\in\mathbb{N}$$ Other than formal manipulation of the Taylor series of the $\sin$ function, is there a way to expand ...
2
votes
2answers
391 views

Given a perturbation of a symmetric matrix, find an expansion for the eigenvalues

Let $A$ be a real, symmetrix $n\times n$ matrix with $n$ distinct, non-zero eigenvalues, and let $V$ be a real, symmetric $n\times n$ matrix. Consider $A_{\varepsilon}=A+\varepsilon V$, a ...
5
votes
4answers
166 views

Compute $\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right )^2+\left (\frac{1}{n} \right )^2$

Compute the value of the following expression $$\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\left ( \frac{1}{2}+\cdots + \frac{1}{n}\right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right ...
2
votes
1answer
90 views

Series $\sum_{n=1}^{+ \infty}\frac{z^{n}}{n}e^{n^{2}z}$

Let $$f(z) = \sum_{n=1}^{+ \infty}\frac{z^{n}}{n}e^{n^{2}z} \ \ \ \ ,z\in \mathbb{C}$$ I want to find the maximal region in which $f$ is holomorphic. I have a problem with the convergence in $\{+i, -i ...
2
votes
1answer
239 views

Formal series expansion of differential operator (d/dx + f(x))^n

My original problem was to find the "coefficient" functions $\varphi_{k,n}(x)$ in $$ (\partial_x + f(x))^np(x) = \sum_{k=0}^n\varphi_{k,n}(x)\partial_x^kp(x). $$ (i.e. find the coefficients ...
1
vote
1answer
131 views

Find the radius of convergence for $\sum^{\infty}_0 n^nz^{n^n}$

Find the radius of convergence for $\sum^{\infty}_{n=0} n^nz^{n^n}$ This is not a power series, but if I define $a_k=k$ if $k=n^n$ and $a_k=0$ otherwise, I would have a power series such that ...
2
votes
1answer
376 views

Reversion of power series

So, I just heard about this method. How does one determine the coefficients, and what is it used for? For example, given $$ y = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7)$$ reversion would give a ...
1
vote
2answers
118 views

Is there a real power series with radius of convergence 1 that converges at 1 but not at -1?

I can find a power series that has radius of convergence 1 but since any series that converges absolutely converges, I cannot find any that converges at 1 but diverges at -1... Can you help me? Thank ...
3
votes
1answer
261 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...