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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0answers
80 views

Upper Bound on $\frac{1}{1-\beta u}-\sum\limits_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$

Is there any procedure to find an upper bound of the following expression? $$\frac{1}{1-\beta u}-\sum_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$$ Here $u,\beta\in\mathbb{R},\ u>1,\ ...
4
votes
2answers
248 views

Taylor series Question

So I have a test next week and I saw this question with no answer and I would like to some help. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ infinitely differentiable and let $\sum _{n=0}^{\infty} ...
1
vote
2answers
101 views

Find the limit of $\lim_{n \to \infty}\frac{(-1)^{n^2}}{3} + 2 - \frac{1}{(-e)^n}$

I want to find the limes of function: $$\lim_{n \to \infty}\frac{(-1)^{n^2}}{3} + 2 - \frac{1}{(-e)^n}$$ If I break up the statement I get $$\lim_{n \to \infty}\frac{(-1)^{n^2}}{3} = \frac {1}{3}$$ ...
0
votes
1answer
84 views

Power series of solution of differential equation around an arbitrary point

How do you determine the series solution to $ y'=y $ or $ y''=-y$ around an arbitrary point, but I would love to see an example around the point 1. I know the solution is $ c_{1}e^{x} $ and $c_{1} ...
0
votes
1answer
81 views

Find the limit $a$ of the sequence $(a_n)_n$

I want to find the limit $a$ of the sequence $(a_n)_n$ and the smallest natural number N such that $|a_n - a| < \epsilon \forall n \geq N$ My $a_n = 5/3 - 3^n/4^n$ for all $n \in \mathbb{N},$ ...
1
vote
2answers
100 views

How to solve $f_{n}(x)=xf_{n-1}(x)-f_{n-2}(x)$?

How to solve the following recurrence relation $$f_{n}(x)=xf_{n-1}(x)-f_{n-2}(x)$$ with the initial conditions $f_{1}(x)=x, f_{2}(x)=x^2-1$? The answer is that ...
4
votes
1answer
113 views

A property of power series and the q-th roots of unity

I'm trying to understand why if $ \displaystyle \sum_{n=0}^{\infty} a_{n}x^{n} = f(x) $, then $$ \sum_{n=0}^{\infty} a_{p+nq} x^{p+nq} = \frac{1}{q} \sum_{j=0}^{q-1} \omega^{-jp} f(\omega^{j} x)$$ ...
2
votes
2answers
1k views

invert a power series?

If one is given a power series of the form: $\sum^\infty_{m=1} a_m x^m = a_0 + a_1x + a_2 x^2 + \dots$ For known $a_m$'s i.e $a_m = f(m)$ and as $m\rightarrow\infty$, $a_m\rightarrow0$ is there ...
1
vote
0answers
61 views

A functional power series equation

I am interested in solving the following functional equation: $$(1-w-zw)F(z,w)+zw^nF(z,w^2)=z-zw\,.$$ Here $n\geq2$ is a fixed integer and $F(z,w)$ is a power series in two variables with complex ...
4
votes
4answers
401 views

Convergence of $\sum_{n=0}^{\infty}\dfrac{z^n}{1+z^{2n}}$

For what complex values of $z$ is $$\sum_{n=0}^{\infty}\dfrac{z^n}{1+z^{2n}}$$ convergent? I would like to write the sum as a power series, because with a power series we can determine the radius ...
3
votes
5answers
102 views

Power series $\sum n^3a_nz^n$

If $f(z)=\sum a_nz^n$, what is $\sum n^3a_nz^n$? The desired sum is $a_1z+8a_2z^2+27a_3z^3+\cdots$. I can't see how to write the desired sum in terms of $f$. For example, I could substitute $kz$ ...
1
vote
1answer
64 views

“Identical” power series

This is a question that came to my mind as I was reading about power series. Is it possible for two power series to, in some sense, represent the same function, and thus be "identical"? So the ...
3
votes
4answers
399 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 $$\dfrac{1}{(1-z)^2} = (1+z+z^2+\ldots)(1+z+z^2+\ldots) = 1+2z+3z^2+\ldots.$$ ...
1
vote
0answers
62 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: $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$ ...
0
votes
1answer
56 views

Simple differentiation question that I am unsure about

I am in the process of re-learning differentiation and am stuck on this as part of a larger problem. Can you explain to me why when differentiated 4 times this: $$y = \sum_{n=0}^{+\infty} ...
1
vote
1answer
151 views

Finding coefficients of a differential equation represented by power series

I am studying for a discrete mathematics exam and have gotten stuck on this question: Any function y of a real variable x that solves the diff erential equation: $$\frac{d^4y}{dx^4} -16y =0$$ may ...
1
vote
1answer
103 views

Show an infinite series of complex number converges

Suppose a sequence of complex number $(z_n)_{n=1}^\infty$. Show that if the series ( of positive numbers) $\sum_{n=1}^\infty {|z_n|^2}$ converges, then the series (of positive numbers) ...
4
votes
1answer
98 views

Calculating $\arctan(3)$ using Taylor series

I'm trying to get a Taylor series equivalent for $\arctan(3)$, but the standard definition for $\arctan(x)$ is restricted to $|x| \le 1$. How can I get a Taylor series for this expression?
2
votes
2answers
161 views

The meaning of Generating functions

I had a look to this video on the field of series and sequences which I know not much about! This guy looked for the generating function of $a_n = 2a_{n-1} + 4a_{n-2}$ with $a_0=1$ and $a_1=3$. The ...
5
votes
0answers
132 views

“Evaluation Homomorphisms” for Formal Power Series

In the ring of formal power series $\Bbb R[[x]]$ it is easy to check by induction that $$ 1 = (1-x)(1 + x + x^2 + \cdots). $$ Does this derivation imply the same identity for those real or complex ...
0
votes
1answer
205 views

Why use series solution rather than variation of parameters?

When should we use series solution to solve a general 2nd order ODE rather than the variation of parameters? Could both methods be used to solve any 2nd order ODE or are there restrictions on when ...
1
vote
0answers
148 views

Power series expansion of the minimum eigenvalue of a linear matrix function

Let $M(\alpha) = A + \alpha \, B$, where A, B are two $n\times n$ positive-semidefinite matrices, and $\alpha$ is a scalar. Define $\lambda(\alpha)$ as the smallest eigenvalue of $M(\alpha)$, i.e., $$ ...
3
votes
2answers
266 views

Is there an explicit formula for power series of $\left(\frac{1+x}{1-x}\right)^n$?

I am trying to answer this question. Per suggestion in one the comments for that question, one might be able use the power series of the terms to arrive at the answer. However, one of the terms is: ...
1
vote
1answer
71 views

Simplification of Binomial Expansion.

How $$(x+h)^n-x^n=nhx^{(n-1)}\text{ ?}$$ My attempt : $$ \begin{align} (x+h)^n-x^n & =nhx^{(n-1)} \\[8pt] & =\left[\sum_{k=0}^{n}\binom{n}{k}x^{(n-k)}h^k\right]-x^n \\[8pt] & = ...
1
vote
1answer
176 views

Applications of higher powers of trigonometric functions

I am after a reference (book, papers etc) about the practical applications of trigonometric functions raised to higher powers. An example is one that I have been using in my own studies: $\cos^4 ...
4
votes
1answer
162 views

The series of $\frac{1}{\cosh(z)}$

How to show that $$\frac{1}{\cosh(z)} =\sum _{n=0}^{\infty }{\frac {\left( -1 \right)^{n}\left(\psi \left( 2\,n,\frac{3}{4}\right)-\psi \left( 2\,n,\frac{1}{4} \right) \right) {z}^{2\,n}}{ ...
10
votes
3answers
289 views

Estimate $\displaystyle\int_0^\infty\frac{t^n}{n!}e^{-e^t}dt$ accurately.

How can I obtain good asymptotics for $$\gamma_n=\displaystyle\int_0^\infty\frac{t^n}{n!}e^{-e^t}dt\text{ ? }$$ [This has been already done] In particular, I would like to obtain asymptotics that ...
2
votes
3answers
168 views

How maths can help to compute convergence?

$r'(\theta)^2 + r(\theta)^2 = \theta^2,\quad r(t=0)=0\tag{1}$ There is an interesting approach to prove that the solutions of the equation $(1)$ have power series representations of form ...
3
votes
0answers
124 views

Changing the power series of sin(x), and its waves are getting bigger. How big do they get?

Motivation: I've been thinking about the transformation of power series, which takes the (power series of) $\exp(x)$ to $\sin(x)$. At first i was trying the series $\sum_{n=0}^{\infty} ...
0
votes
1answer
105 views

Series Expansion of an Exponential with a Trig Function in the Exponent

Can anyone get a general expression for $$e^{a\cos x}$$ in terms of an infinite sum? I'm having trouble with a general form in terms of $n$ for the coefficients... Alex
1
vote
2answers
55 views

General case of radius of convergence of a power series

Show that if the series $\sum_{n=1}^{\infty} a_nx^n$ has a radius of convergence $L = R$ so the series $\sum_{n=1}^{\infty} a_nx^{kn}$ has radius of convergence $L = R^{\frac{1}{k}}$. Anyone could ...
0
votes
2answers
133 views

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$ [duplicate]

Show that $\sum_{n = 1}^{+\infty} \frac{n}{2^n} = 2$. I have no idea to solve this problem. Anyone could help me?
1
vote
1answer
67 views

Can a non-zero power series be zero on some interval inside the radius of convergence?

Suppose there is a real sequence $(c_i)_{i=0}^\infty$ such that $\sum_{i=0}^\infty c_ix^i$ converges everywhere on some interval (a,b). Let (s,t) be an interval such that $(s,t)\subset(a,b)$ and such ...
1
vote
2answers
47 views

Let $\sum_{n=1}^{\infty}n^5(\frac{x}{x+2})^n=S(x)$. Prove that the sum S(x) is a function and continuous to $x\epsilon [0,10]$

Let $\sum_{n=1}^{\infty}n^5(\frac{x}{x+2})^n=S(x)$. Prove that the sum S(x) is a function and continuous to $x\epsilon [0,10]$ Since we are talking about sums and we need to prove continuous i ...
1
vote
1answer
122 views

Sum of power series with inverse coefficients

Are there any relations between the sums of $\displaystyle\sum_{n=0}^{\infty} a_n x^n$ and $\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{a_n}$? For example, I know ...
1
vote
0answers
97 views

$\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}x^k =$

Prove that for $|x|<1$, $\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}(-x^2)^k = ...
0
votes
1answer
97 views

A problem about the convergence of a power series

Let $\{a_n\}_{n\in\mathbb N}$ be a real sequence with $a_n\ge0$ for all $n\in \mathbb N$. If $$\sum_{n=1}^{\infty}a_n<\infty$$ I have to discuss the radius of convergence (in $\mathbb R$) of ...
5
votes
1answer
374 views

Isomorphism of formal power series factorrings over polynomials

This problem is taken from the Hartshorne's book Algebraic Geometry, Chapter 1, Section 5, Problem 14(a). Two polynomials $f(x,y)$ and $g(x,y)$ are written in the form $$f(x,y) = f_{r}(x,y) + ...
0
votes
1answer
80 views

What's Taylor expansion of: $f(x)=\frac 1x\ln{(1+2x^2)}$?

What's Taylor development on the next function: $f(x)=\frac 1x\ln{(1+2x^2)}$? Actually this one is the first question I've seen with $ln$, My instincts tell me to try and do derivative in order to ...
1
vote
0answers
91 views

Series expansion of $\frac{1}{(1-x)^{-n}-1}$

Let $f_n(x)=\frac{1}{(1-x)^{-n}-1}$ with a positive integer $n$. I am interested in upper-bounding this function for small positive values of $x$ in terms of $x$ and $n$. I don't think that the ...
0
votes
2answers
68 views

proof of $\sum_{n=1}^\infty n \cdot x^n= \frac{x}{(x-1)^2}$ [duplicate]

I know that the Series $\sum_{n=1}^\infty n \cdot x^n $ converges to $\frac{x}{(x-1)^2}$ but I'm not sure how to show it. I'm pretty sure that has been asked before, but I wasn't able to find ...
4
votes
2answers
397 views

Formal (series/sum/derivative…)

I have come across a lot of cases where terms such as formal sum rather than simply sum is used, similarly in case of derivatives/infinite series/power series. As I understand in case of series/sum, ...
2
votes
1answer
94 views

Interval of convergence $\sum_1^\infty \frac{2^n}{3n}(x+3)^n$

$$\sum_1^\infty \frac{2^n}{3n}(x+3)^n$$ I do the ratio test to find the radius. $$\frac{2^{n+1}}{3(n+1)}(x+3)^{n+1} *\frac{3n}{2^n (x+3)^n}$$ This should reduce down to $2|x+6|< 1$ This is ...
1
vote
1answer
105 views

Taylor series of a power series.

Consider a power series $f(x)$ around a point $c \neq 0$. Then is it equal to its Taylor series around $0$? The reason I am wondering about this is because if it is true even for some special cases, ...
0
votes
1answer
70 views

Interval of convergance $\sum_0^\infty \frac{(2n)!}{(n!)^3}*x^n$

$$\sum_0^\infty \frac{(2n)!}{(n!)^3}*x^n$$ I have no idea how to do this. I tried to write it all out with the ratio test and I get some weird expression that doesn't make sense like $$x * ...
1
vote
0answers
64 views

Is there such a thing as a removable singularity for a power series on the edge of the convergence disc?

It may be a loss in translation, but I have been taught that a removable (effaçable in French) singularity for a power series lies necessarily within the interior of the convergence disc, yet I found ...
2
votes
4answers
102 views

Power series for $(1+x^3)^{-4}$

I am trying to find the power series for the sum $(1+x^3)^{-4}$ but I am not sure how to find it. Here is some work: $$(1+x^3)^{-4} = \frac{1}{(1+x^3)^{4}} = \left(\frac{1}{1+x^3}\right)^4 = ...
4
votes
1answer
161 views

When does an analytic function grow faster than a polynomial?

Suppose $f$ is an analytic function with power series expansion $f(z)=\sum_{n=0}^{\infty} a_nz^n$, and $p = \sum_{n=0}^{d}b_nz^n$ is a polynomial. If $f$ is a polynomial of degree larger than $d$, ...
-2
votes
1answer
87 views

domain of convergence of power series

I need to find radius of convergences and domain of convergence for the: $\sum_{n=1}^{\infty}3^{n^{2}}x^{n^{2}}$. Can you help me please?
2
votes
2answers
747 views

Induction Proof for a series expansion of a function

I have done induction proofs of many different types, but trying to prove by induction that a derivative from the Taylor series expansion of a function has me stumped in terms of how to get the final ...