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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0answers
16 views

On a certain inequality for the power sum

Let $a_k \ge 0$ and $b_k \ge 0$ for $ k \in N$ such that the following two conditions (i) $0 \le \sum_{k=1}^{\infty}\frac{a_k-b_k}{k^{\alpha}}$ for $0<\alpha \le \alpha_0 <1$, (ii) $ \...
1
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3answers
748 views

Prove sum of $\cos(\pi/11)+\cos(3\pi/11)+…+\cos(9\pi/11)=1/2$ using Euler's formula

Prove that $$\cos(\pi/11)+\cos(3\pi/11)+\cos(5\pi/11)+\cos(7\pi/11)+\cos(9\pi/11)=1/2$$ using Euler's formula. Everything I tried has failed so far. Here is one thing I tried, but obviously didn't ...
0
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2answers
31 views

Power Series Solution for Differential Equation of Shifted Exponential Function

I am trying to write a shifted exponential function as a power series. I am aware of the power series definition of the exponential function, i.e. $e^{-x}=\sum_{n=0}^{\infty}\frac{(-x)^{n}}{n!}$. ...
1
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0answers
38 views

Deciphering the theorem of perfect powers

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets a and c not equal to zero....
2
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1answer
41 views

power series representation of a function by differentiation

Find power series representation for $\frac{1}{(7+x)^2}$ What I tried... $$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$$ $$\frac{1}{7(1-(-{x \over7}))}=\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{7^{n+1}}$$ $${d\...
-4
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1answer
62 views

help in real analysis [on hold]

How can I use this definition to prove that $a^{\frac{1}{n}}$ converge to $1$? where a >0
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1answer
44 views

Order of summation for shifted exponential function

I want to represent the function: \begin{equation} f(x)=e^{-a(x-b)^{2}} \end{equation} where, $0<a<1$, $x\in\mathbb{R}$, and $b\in\mathbb{R}$. As a power series for an integral I am working ...
-1
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2answers
73 views

Reciprocal of the sum of powers of $1/x$ [duplicate]

Incidentally, I found $$\frac{1}{\sum_{n=1} \frac{1}{x^{n}}} = (x-1)$$ where $x\ge 2$. Please direct me to how others have developed the relationship. My computer cannot compute more than X = ...
3
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2answers
48 views

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$.

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$. Find a closed form expression for all x which converge and hence evaluate $\sum_{n=1}^\infty\frac{(-1)^{n-1}}{(2n+...
4
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0answers
101 views

Integral of combination of power, exponential, and confluent hypergeometric function

I am trying to solve a couple integrals of the form: \begin{equation} \int_{0}^{\infty} x \, e^{-a(gx-b)^{2}}\,e^{-\beta_{1}x}\, {_{1}}F_{1}(-\alpha_{1};-\alpha_{3};\beta_{3} x) \ \mathrm{d}x \end{...
1
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1answer
76 views

Are all the zeros of $1-a_2x^2+a_4x^4-a_6x^6+\cdots$ real for $a_{2n}>a_{2(n+1)}$ with $a_{2n+1}=0$ and $a_{2n}>0$?

This question is related to a previous question of mine. I was not pleased about the conditions I provided there. I had something different in mind but I failed in stating it. So here are the ...
5
votes
2answers
96 views

Series of Functions and Continuity

Let $a > 0$, and $(f_n)_{n=0}^{\infty}$ a sequence of continuous functions $f_n:[-a,a] \rightarrow \mathbb{R}$. Assume that the series \begin{equation} \sum_{n=0}^{\infty} x^n f_n(t) \end{equation} ...
0
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0answers
41 views

what is the asymptotic expansion of this $_2F_2$ function?

We need to expand the function $_2F_2(a+b x,1; 1+a+b x, b x; x)$ near $x=+\infty$. Where $a$ is complex, $b>1$. When $x\to+\infty$, both the parameters and the variable goes to infinity, we can ...
1
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0answers
89 views

Ellipsoidal harmonics - A Series expansion for Lame functions of the second kind

Intro to skip In the theory of ellipsoidal harmonics, Lame functions of the second kind $F_n$ arise as the second linearly independent solution (the first being Lame functions of the first kind $E_n$)...
3
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1answer
27 views

Find the radius of convergence of a series containing odd and even terms separately.

Given series $\sum_{n=0}^{\infty}a_nz^n$ where $a_n=\begin{cases}\dfrac{1}{3^n} & \text{when $n$ is even} \\ \dfrac{1}{5^n} & \text{when $n$ is odd} \end{cases}$ Find the radius of ...
1
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1answer
25 views

Find a power series representation for the function and determine the interval of convergence

Find a power series representation for the function $f\left( x \right)=\frac { x }{ 2{ x }^{ 2 }+1 } $ and determine of convergence. I ended up with the following: $$\sum_{n=0}^{\infty} (-2)^n x^{...
3
votes
4answers
317 views

Power series with alternating signs and its zeros.

This may be a strange question, but I've not found anything about this. Well, anyone can observe that both $$ \cos(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}z^{2n} $$ and $$ \sin(z)=\sum_{n=0}^{\...
3
votes
1answer
48 views

Power Series in Two Variables and Radius of Convergence

Let $\alpha > 0$, $\beta > 0$, and assume that the power series with real coefficients \begin{equation} \sum_{n,m = 0}^{\infty} a_{n,m} x^{n} y^{m} \end{equation} is absolutely convergent for ...
2
votes
3answers
64 views

Why does changing the center of a geometric power series change the interval of convergence?

I know that the interval of convergence of the geometric power series $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$ is $(-1,1)$. Why is it that if I do the following manipulation $$\frac{1}{1-x}=\frac{1}{...
1
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0answers
31 views

Question About Cauchy Product;

Is there a formula to multiply many series (More than two) using the Cauchy product? If there isn't, please tell me how I can write this formula $ \left( \frac{1}{a-e^x} \right) ^{n+1}$as the ...
1
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0answers
41 views

Show that $\sum (-1)^n x^{(2^n)}$ has no limit as $x \uparrow 1$

Show that the following limit does not exist: $$\sum_{0}^{\infty} (-1)^n x^{(2^n)}\text{ with }x \uparrow 1$$ I tried setting $$f(x) = x - x^2 + x^4 - x^8...$$ then $$f(x) = x-f(x^2)$$ then the ...
195
votes
17answers
14k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
1
vote
1answer
100 views

Prove that $\displaystyle\int_0^\infty \frac{\sin x}{x}dx$ converges using power series

$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-...=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!}$$ $$\frac{sin x}{x} = 1-\frac{x^2}{3!}+\frac{x^4}{5!}-...=\displaystyle\sum_{n=0}^\infty(-1)^...
0
votes
1answer
26 views

Convergence radius for power series $ y(x)=\sum^{\infty}_0 a_nx^n\;$ with recurrence relation [on hold]

Suppose the power series $ y=\sum^{\infty}_0 a_nx^n\;$ has terms that satisfies the recurrence relation: $$a_{n+2}=\frac{(2n_1)(3n+2)}{(n+3)(2n-5)}a_n$$ with $a_0=1$,$a_1=0$. What is the radius of ...
7
votes
2answers
112 views

Stationary distribution of “probabilistic geometric series” with two alternative ratios

I have an iterative process starting at $X_0=2$. In each iteration $i=1,2,\ldots$, the value of $X_i$ is determined based on the value of $X_{i-1}$ as follows: With probability 0.5, $X_i=qX_{i-1}+1$, ...
3
votes
2answers
48 views

Analytic continuation of $\sum (z/a)^n$

I'm having trouble continuing this function beyond its convergence radius, $R=a$. $$f(z)=\sum (z/a)^n$$ Given the context (a textbook in complex analysis) I suspect it should have a simple closed-...
0
votes
3answers
28 views

Find a power series representation for this function

Find a power series representation centered at $x=0$ for $f(x) = \dfrac x {15x^2+1}$. Answer: $f(x) = \sum \limits _{n=0} ^\infty (-1)^n (15x^2)^{n+1}$. For some reason this just isn't clicking ...
1
vote
1answer
52 views

Analytic continuation of $\sum z^n/n$

I have the following function $$f(z)=\sum_{n=1}^\infty \frac{1}{n}z^n$$ It is easy to see that it converges for $|z|<1$ (root test, for example). How can it be analytically continuated beyond ...
2
votes
1answer
49 views

Radius and Interval of Convergence $\sum_{n=0}^{\infty}\frac{7^n}{n!}x^n$

$\displaystyle \sum_{n=0}^{\infty}\frac{7^n}{n!}x^n$ I'm still trying to get the hang of these and feel like I've done something wrong here. After applying the ratio test I end up with: $\left|7x\...
2
votes
3answers
55 views

Find a power series representation for the function. [closed]

I'm not sure how to handle this problem. I got that the radius of convergence was 1/6, but I don't know how to represent the function as a power series. I can modify it to look like the following: x ...
3
votes
0answers
32 views

Question on possible Bilinearity of the Action of a Linear Functional on a Polynomial

I have been self studying the classical Umbral Calculus and have been reading works and papers from Rota and Roman on the material and I have a question regarding the following. The text uses $$\...
0
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1answer
44 views

Tight bounds on the sum of the $j$th power of the first $n$ natural number?

What are the tight bounds for $S_{n,j}=\sum_{k=1}^n k^j$? Where $O(j)=O(n^3)$.
2
votes
2answers
55 views

Problem calculating the sine of a matrix

Given the matrix $A=\begin{pmatrix}-\frac{3\pi}{4} & \frac{\pi}{2}\\\frac{\pi}{2}&0\end{pmatrix}$, I want to calculate the sine $\sin(A)$. I do so by diagonalizing A and plugging it in the ...
2
votes
2answers
121 views

Is it possible to have a power series for arctan(x) centered at 1?

My Calculus 2 professor referenced that such a series is impossible, but why? I understand how to properly find the power series of arctan(x) centered at 0.
3
votes
1answer
51 views

How to determine sum of an alternating power series and to prove that sum is positive

I am working on a problem involving an alternating power series as follows: $$\sum_{i=0}^{a-2} (-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}$$ $a$ and $b$ is constant with $0<x<1$ I would like to ...
1
vote
0answers
29 views

Quotient of Confluent Hypergeometric Functions of the 1st Kind

I want to solve the following problem for x: \begin{equation} \frac{\mathrm{d}}{\mathrm{d}x}\ e^{-\beta_{1}x}\,{_{1}}F_{1}[-\alpha_{1};-\alpha_{3};\beta_{3}x]=0 \end{equation} where, $\alpha_{1},\...
0
votes
1answer
22 views

Radius of convergence of integral series; problem with limsup

Let $\sum c_k x^k$ be a power series with radius of convergence $R$. Then the integral series $$\sum_{k=0}^\infty \frac{c_k}{k + 1}x^{k+1}$$ also has radius of convergence $R$. I'm reading Real ...
-1
votes
3answers
34 views

Find the radius of convergence as well as the interval of convergence:

I've done a bunch of these and was successful, but this one is proving to be troubling. I have no idea how to handle that n*sqrt(n) at the bottom. I ended up with some jank shit like this lol: Lim ...
0
votes
1answer
20 views

Find the relative width of a guitar fret

There is an equation to find the position of a fret on a guitar fretboard, given the length of a string is given by \begin{eqnarray} d = s – \frac{s}{2 ^ {(n / 12)}}, \end{eqnarray} where $d$ is the ...
0
votes
2answers
27 views

Why is $\operatorname{Sech}(x)$ Taylor series divergent past $\pi/2$?

Someone asked a question here about why the Taylor series of $\log(1+x)$ diverges: Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$? I have a similar question: why ...
-1
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1answer
65 views
1
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2answers
28 views

Find the Taylor-series expansion of a square of a rational function of a complex variable

I've been trying to find the Taylor-series expansion of the following function: $$ f(z)=\left ( \frac{1+z}{1-z} \right )^2 $$ az the origin : Z0 = 0. also I would like to find the region of ...
1
vote
1answer
69 views

What is the interval of convergence of : $\sum_{n=1}^\infty\frac{n^n}{n!}x^n$?

$x+ \frac{2^2x^2}{2!}+ \frac{3^3x^3}{3!}+ \frac{4^4x^4}{4!}+...$ Possible answers- 1.($0,1/e$) 2.(1/e, $\infty$) 3.(2/e, 3/e) 4.(3/e, 4/e) ...
3
votes
1answer
25 views

Field and ideal notation: double bracks/parens vs single brackets/parens

I'm reading some notes that has the following denotation: the set of formal power-series with coefficients in $\mathbb{F}_p$ is denoted by $\mathbb{F}_p[[t]]$. the fraction field, $\operatorname{...
0
votes
0answers
14 views

Sum of a converging series having Error function with a polynomial

I am struggling to find the sum of the following series: $k\sum\limits_{z=1}^{\infty} \frac{(z+1)^2}{4} . erfc(az)$ where $k$ and $a$ are known parameters and $erfc(x)$ is the complementary error ...
2
votes
5answers
1k views

What does it mean intuitively for a Taylor Series to be centered at a specific point?

I understand what a Taylor series is and how to find the Taylor series of a function. However I do not understand intuitively what it means to find a Taylor series for a specific function, centered at ...
1
vote
3answers
113 views

Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation

Find sum of the series $$\sum_{n=1}^{\infty}\frac{x^{3n}}{\left(3n\right)!}$$ using differentiation. So far I found that $$S(x)+1=S'''(x)$$ but it does not help. Then I tried different interesting ...
4
votes
1answer
207 views

New series formula for $\arctan(x)$?

I discovered this equation, but have no idea if it has been previously discovered. Please help determine if it has been previously developed. Or please prove that the equation is not correct. $$\...
1
vote
1answer
16 views

Transition from convolution of PMF's to convolution of power series in a random walk

In the proof that symmetric random walks end up regressing to the origin with probability $1$, I have found this didactic post on-line. In it the following two definitions are given: Probability of ...
13
votes
4answers
500 views

How can we show that $ \sum_{n=0}^{\infty}\frac{2^nn[n(\pi^3+1)+\pi^2](n^2+n-1)}{(2n+1)(2n+3){2n \choose n}}=1+\pi+\pi^2+\pi^3+\pi^4 ?$

We proposed this sum, but we are lacking in knowledge of this area of maths and we would ask if any of the authors would be willing to show us step by step how to go about proving this sum. $$ \sum_{n=...