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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2answers
64 views

Computing $\displaystyle \sum_{k\ge2}k(1-p)^{k-2}$

Computing $\displaystyle \sum_{k\ge2}k(1-p)^{k-2}$, $p\in ]0,\space1[$ WolframAlpha says it is $\cfrac {p+1}{p^2}$ but I couldn't get that value but anyway here is what I did: $$\displaystyle ...
3
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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 ...
1
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1answer
103 views

Do $\sum_{n=0}^{\infty}\frac{1}{a_{n}}$ and $\sum_{n=0}^{\infty}\frac{a_{n-1}}{a_{n}}$ and $\sum_{n=0}^{\infty}e^{-a_{n}}$MUST converge?

Let $\left\{a_{n}\right\}$ be a strictly increasing sequence of positive numbers, Do $$\sum_{n=0}^{\infty}\frac{1}{a_{n}}$$ and $$\sum_{n=0}^{\infty}\frac{a_{n-1}}{a_{n}}$$ and ...
7
votes
1answer
220 views

Infinite series where each term is the square of the last

Is there a closed-form, in terms of elementary functions or otherwise, for the power series $x+x^2+x^4+x^8+x^{16}+...$, where each term is the square of the last?
3
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1answer
283 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
4answers
173 views

Index change of variable with product

This is probably a dumb question, but oh well here it is: One day a student came to me and asked: Why isn't $$ \sum_{k=0}^{\infty} \frac{1}{2^{2k}}=\sum_{n=0}^{\infty} \frac{1}{2^{n}}$$ using a ...
2
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5answers
292 views

Identify infinite sum: $\sum\limits_{n=0}^{+\infty}\frac{x^{4n}}{(4n)!}$

Find $f(x)$, the unknown function satisfying $$f(x) = \sum\limits_{n=0}^{+\infty}\frac{x^{4n}}{(4n)!}$$ I'm looking for a direct solution which is different from mine, if possible.
0
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1answer
387 views

Try to find an approximation by logarithm function.

Recently I am thinking about this question: Assume $x$ is real, $x\geq0$, $c$ is a positive constant number and $z$ is also a real constant between $3.5$ and $4$. Now there is a function: $$ ...
2
votes
2answers
158 views

The ring of formal power series

Is there a simple proof/clarification of this statement? The set of all formal power series in X with coefficients in a commutative ring R form another ring that is written R[[X]], and called the ...
0
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1answer
93 views

Evaluate series sum

Let $\alpha$ be some positive noninteger real constant and $n$ be an arbitrary nonnegative integer. Consider a series $$ S_{n}(x) = \sum\limits_{k=0}^{\infty} {\alpha \choose k} ...
21
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3answers
504 views

Math contest: Find number of roots of $F(x)=\frac{n}{2}$ involving a strange integral.

Edit summary: A good answer appeared. CW full answer added, based on given answers. Removing my ugly-looking attempts, as they still remain in the rev. history. Here's a final-round calculus ...
0
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2answers
110 views

Where's the singular point of the series $\sum_{v\ge 1}v^{-2}z^v$?

The radius of convergence of the series $\sum_{v\ge 1}v^{-2}z^v$ is 1, so there must be a singular point on the boundary. But for every $|\zeta|=1$, $|\sum_{v\ge 1}v^{-2}\zeta^v|\le\sum_{v\ge ...
0
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2answers
252 views

Radius of Convergence of “Shifted” Power Series

Suppose that $\sum_0^\infty a_nz^n$ has radius of convergence $1$ and suppose that $|z_0|=r<R$ Let $g(z)=\sum_0^\infty a_n (z-z_0)^n$ Prove that $g(z)$ has radius of convergence at least $R-r$. I ...
13
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1answer
556 views

Does a power series vanish on the circle of convergence imply that the power series equals to zero?

Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be a power series, $a_n, z\in \mathbb{C}$. Suppose the radius of convergence of $f$ is $1$, and $f$ is convergent at every point of the unit circle. Question:If ...
2
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2answers
205 views

Approach to limit of infinite product

I was wondering if there is any proof that the limit of infinite product $$\lim_{n \to \infty} \prod_{i=1}^{n} x_i, \mathrm{where}$$ $$0 < x_i < 1$$ is equal to 0 and that it does not ...
1
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1answer
398 views

Error bound for Taylor approximation to $\tan(x)$

I'm looking to bound the error for the Taylor series of $\tan(x)$ so that I will know how many terms I need to go out to get a desired precision. I've already searched and came across this, but the ...
2
votes
1answer
127 views

What are the subsets of the unit circle that can be the points in which a power series is convergent?

Let $A\subset\Bbb C$ be a subset of the unit circle. Consider the following condition on $A$. Cond. There exists a sequence $\{a_i\}_{i=1}^\infty$ of complex numbers such that $$\sum_{n=1}^\infty ...
5
votes
2answers
605 views

Singularities of $e^{z - \frac{1}{z}}$

I believe $e^{z - \frac{1}{z}}$ has essential singularities at $z = 0$ and $z = \infty$ (in both cases because of a $\frac{1}{z}$ in the exponent) but I'm having a hard time proving this. How can one ...
4
votes
1answer
246 views

Definite integral of tetration between $0$ and $1$

In my old writes I found next formula, where is ${_{}^2}x$ is tetration: $$\int_0^1 {_{}^2}x \ dx = \sum\limits_{i=1}^\infty \frac {(-1)^{i+1}} {{_{}^2}i} \approx 0.783430511\ldots$$ And now I am ...
3
votes
3answers
245 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 $0<q<1$.
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2answers
514 views

Why do we need Taylor polynomials?

This question doubles as "Is my understanding of what a Taylor polynomial is for, correct?" but In order to write out a Taylor polynomial for a function, which we will use to approximate said function ...
12
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1answer
317 views

Solving a formal power series equation

I want to find a function $f(x,y)$ which can satisfy the following equation, $$\prod _{n=1} ^{\infty} \frac{1+x^n}{(1-x^{n/2}y^{n/2})(1-x^{n/2}y^{-n/2})} = \exp \left[ \sum _{n=1} ^\infty ...
0
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0answers
83 views

Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $ Is it related to a known transcendental function? From the research I did, it ...
3
votes
0answers
256 views

Chebyshev Diff EQ

Find a power series solution about $x_0=0$ for the Chebyshev differential equation $$(1-x^2)y''-xy'+n^2 y=0,$$ as a function of of the integer $n$. Show that the solutions form a terminating ...
0
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1answer
225 views

find the power series of the problem…

""Find a power series associated with the problem where we have to find a number of ways to select 10 people to form and expert committee from 6 Professors and 12 Associate Professors."" Question ...
0
votes
1answer
204 views

Limit of the determinant of a series of matrices

Given the $N\times N$ matrix $A$, consider the series: $$B=\sum_{k=1}^{N}(A^k)^{-1}$$ where the symbol $o^{-1}$ means the inverse of $A^k$ is it possible and if yes how, to find all the matrices for ...
0
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1answer
121 views

Power-Series question

Suppose, $ P(x) := \sum_{n\geqslant r} {p_{n}x^n}$ and $ Q(x) := \sum_{n\geqslant s} {q_{n}x^n} $ are formal power series, where $p_{r}$ and $q_{s}$ doesnt equal $0$ such that $x^r$ is the ...
3
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2answers
446 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 $\forall x \in \mathbb{R}-\{1\}$. ...
-2
votes
1answer
153 views

Radius of convergence of a series of random variables

Let $X_n$ be i.i.d. and (a.s.) bounded random variables.(none of them identically zero) Prove that the radius of convergence of the series with coefficients $X_n$, ...
6
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1answer
902 views

prove that the entire function f is a polynomial.

Suppose that $f$ is an entire function, and that in every power series $f(z)=\sum_{n=0}^{\infty} c_{n}(z-a)^n$ at least one coefficient is 0. Prove that $f$ is a polnomial. Hint: ...
0
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1answer
142 views

Is there any “nice” function whose MacLaurin series has certain properties?

In learning about asymptotic expansions of functions, I've encountered several problems where a particular pattern of powers is coming into play, and I'm finding functions that I can readily show to ...
2
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1answer
134 views

How can I express such function as known functions or power series?

$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)$$ $$\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$ ...
3
votes
1answer
160 views

Is the set of all power series with convergence radius equal to $1$ a vector space?

Given the set of all power series with radius of convergence ($r$ in the definition) equal to one: $$A:=\{\sum a_kz^k | r =1\}$$ Does $A$ form a vector space? The radius of convergence doesn't ...
1
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0answers
45 views

Power series over integral change of variables

Let $y$ be given by $$y=\int_x^\infty \frac{dx'}{(1+x'^2)^\alpha}$$ where $\alpha>1$. Is it possible to express the following as a series: $$x(1+x^2)^{\alpha-1}=\sum a_n\left(\frac{1}{y}\right)^n$$ ...
0
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2answers
279 views

Behavior of $\sum_{n=1}^\infty n^{-1}z^n$ on the circle of convergence

Consider the following complex power series :$$\sum_{n=1}^\infty\frac{z^n}{n}$$ The radius of convergence of this series is $1$ and the series is divergent for $z=1$. I want to know what are the ...
1
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2answers
113 views

Need help calculating $\sum_{i=0}^\infty \dfrac{e^{-\lambda}\lambda^{2i}}{(2i)!}, 0 \lt\lambda \lt 1$

I am trying to calculate the following sum: $$\sum_{i=0}^\infty \dfrac{e^{-\lambda}\lambda^{2i}}{(2i)!}, 0 \lt\lambda \lt 1$$ It is clearly a power series, however I am stuck trying to move beyond ...
1
vote
1answer
191 views

simple pendulum

I have to deal with this integral in order to compute the period of a pendulum $$ \int^{\theta_{0}}_{0}\frac{d\theta}{\sqrt{\cos\theta_{0}-\cos\theta}} $$ I was asked by my instructor to solve this ...
1
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0answers
96 views

generalizing superfunctions of entire functions

Let $z$ be complex and $x$ real. Define $f(z,0) = f(z)$ where $f(z)$ is an entire function. Define $f(z,x)$ as the $x$ th superfunction of $f(z)$. We know that $f(z,x-1) = f(f^{-1}(z,x)+1)$ where ...
1
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1answer
347 views

Is convergence of a sequence of power-series equivalent to the convergence of the respective coefficients?

Show that convergence of a sequence of power-series is equivalent to convergence of their respective coefficients. More precisely, let $\mathbf{a}:=\left[a_{i,j}\right]_{i,j\in\mathbb{N_0}}$ be a ...
3
votes
1answer
133 views

Uniqueness of power series determined by a sequence of points converging to a boundary point

Prove or disprove: A power series that converges on $\left[0,1\right]$ is uniquely determined by any sequence of points converging to $1$. More precisely, let $f, ...
0
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1answer
111 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 ...
2
votes
1answer
220 views

Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?

Let $x$ be a positive real number. Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$. Call this function $f(n,x)$. Can we give good upper and lower bounds of $f(n,x)$ ...
3
votes
1answer
119 views

Formal exponential of multivariate power series

I was wondering about this. Consider a formal power series $$\sum_{n=1}^{\infty} a_n x^n$$. We can find its formal exponential, given by $$\exp\left(\sum_{n=1}^{\infty} a_n x^n\right) = ...
0
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1answer
156 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 $i>0$. ...
1
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1answer
265 views

Power series for the sawtooth wave

This wikipedia article described a Fourier expansion of the sawtooth wave. Does this wave have a power series expansion (around any point)? If so, what is it? Does every function with a Fourier ...
0
votes
1answer
89 views

Calculating which $(2^x,2^y)$ coordinates intersect diagonally ($45^{\circ}$) with an arbitrary rectangle in 2D space.

Calculating which $(2^x,2^y)$ coordinates intersect diagonally ($45^{\circ}$) with an arbitrary rectangle in 2D space. The rectangle in this case is your computer monitor. Consider navigating a 2D ...
1
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1answer
433 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} & ...
7
votes
1answer
228 views

Closed form for the series $\sum\limits_{k=0}^\infty \frac{(-1)^k\exp(-\lambda(2k+1)^2)}{(2k+1)^3}$

Does there exist an explicit expression for $$\sum_{k=0}^{\infty }\frac{\left( -1\right) ^{k}e^{-\lambda \left( 2k+1\right) ^{2}}}{\left( 2k+1\right) ^{3}}\;,$$ where $\lambda$ is a positive scalar? ...
3
votes
3answers
244 views

Is this sine series bounded?

Let $$f(x)=\sum_{k=1}^{\infty}\sin\left(\frac{x}{2^k}\right)$$ Is $f(x)$ bounded? EDIT: I'm asking if there is $M \in R$ (M constant), s.t. $\forall x $ $|f(x)|<M$ EDIT2: I deleted parts ...
1
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2answers
129 views

Power series: Is the radius of convergence $\frac{1}{3}$ for $ \frac{x}{1\cdot3} + \frac{x^2}{2\cdot3^2} + \frac{x^3}{3\cdot3^3}+…$

I should calculate the radius of convergenc and would like to know, if the result $\frac{1}{3}$ is correct. Here the exercise: $$ \frac{x}{1\cdot3} + \frac{x^2}{2\cdot3^2} + \frac{x^3}{3\cdot3^3} ...