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).

learn more… | top users | synonyms

3
votes
2answers
146 views

How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
3
votes
4answers
223 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
809 views

Proof of the “Radius of Convergence Theorem”

I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
3
votes
1answer
1k 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 ...
3
votes
2answers
391 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
3answers
184 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
369 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
7answers
325 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
2answers
857 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
51 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?
2
votes
3answers
51 views

A power series from $\frac{x}{9+x^2}$

I need to make power series from $\frac{x}{9+x^2}$, and I don't have any idea how. The only thing I know is how to make power series from $\frac{1}{1-q}$. Thank you!
2
votes
0answers
86 views

Irreducible polynomials as formal power series

I'm studing the ring of formal series with complex coefficients $\mathbb{C}[[x]]$. I proved that the polynomial $y^2-x^3-x^2$ is irreducible in $\mathbb{C}[x,y]$ but reducible in $\mathbb{C}[[x,y]]$. ...
2
votes
4answers
169 views

Expansion of $(1-z)^{-m}$

Expand $(1-z)^{-m}$, $m$ a positive integer, in powers of $z$. Since $\dfrac{1}{1-z}=1+z+z^2+\ldots$, we can find $$\dfrac{1}{(1-z)^2} = (1+z+z^2+\ldots)(1+z+z^2+\ldots) = 1+2z+3z^2+\ldots.$$ ...
2
votes
2answers
73 views

Understanding Power Series Multiplication Step

Working on Spivak's Calculus problems, I searched online, trying to understand the solution provided for Problem 4a of Chapter 2. I found the question I needed: Spivak's Calculus - Exercise 4.a of ...
2
votes
1answer
191 views

Definite Sum of Confluent Hypergeometric involving power function

I find it difficult to evaluate the following definite sum: $$ \sum _{k=1}^K \frac{_1F_1[k,1,x]} {2^k} $$ Thank you for your time
2
votes
3answers
108 views

Explicit formula for the series $ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $

I was wondering if there is an explicit formulation for the series $$ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $$ It is evident that the converges for any $x \in \mathbb{R}$. Any ideas on a formula?
2
votes
2answers
68 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
394 views

Power Series With Bernoulli Numbers

The exercise reads "Express the power series for $\large \frac{z}{\sin (z)} = \frac{2 i z}{e^{iz} - e^{-iz}} $ in terms of Bernoulli numbers." I am given in a previous exercise that the Bernoulli ...
2
votes
1answer
1k 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
3answers
85 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
1answer
71 views

Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
1
vote
1answer
149 views

Introduction to Analysis: Multiplication Theorem for Series

I've been stuck on this problem over the weekend so I decided to ask for some direction. The problem reads: "The multiplication theorem for series requires that the two series be absolutely ...
1
vote
0answers
60 views

Solving ODE with negative expansion power series [duplicate]

I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this: $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$ ...
1
vote
1answer
69 views

Modelling ellipsoid-like surface with equations

In my numerical simulation, I would often have to generate the failure surface of some materials. Often, the failure surface will take a ellipsoid-like ball such as this: Of course, the surface ...
1
vote
2answers
131 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
1answer
184 views

using power series expansion to find a holomorphic function which solves a differential equation

Using power series expansions, find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$} and solves the differential equation $(1-z^2)f''(z)-4zf'(z)-2f(z)=0$ for ...
1
vote
0answers
74 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
110 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
64 views

Calculate the Radius of convergence of $\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$

I need your help: Calculate the Radius of convergence of the following: $$ \sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$$ Im new to this subject, so I'd appreciate it if you can add explanations to ...
0
votes
4answers
70 views

Power Series Proof w/ Binomial Coef.

Prove that, for any positive integer k, $$\sum_{n=0}^\infty {{n+k \choose k}z^n}=\frac{1}{(1-z)^{k+1}}, |z| < 1$$
0
votes
1answer
176 views

Uniform Convergence: Complex Analysis

To show that $f_k(z) = \frac{z^k}{k}$ converges uniformly for $|z| < 1 $ and that $f'_k(z)$ does not converge uniformly for $|z| < 1$, what must be done? What other things can be said about the ...
0
votes
3answers
689 views

Approximating cube root function for small values of $x$

How can one show that for small values of $x$, $\sqrt[3]{x+1}\approx1+\frac{x}{3}$?
0
votes
1answer
109 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
546 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
89 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
87 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 ...