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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1answer
23 views

Uniform convergence of the series $\sum_{n=1}^{\infty} \frac{\cos(2nt)}{4 n^2 - 1} $

I am trying to find if this series is uniformly convergent: $$\sum_{n=1}^{\infty} \frac{\cos(2nt)}{4 n^2 - 1} $$ So far I have (using the Weierstrass M-Test): $$| \frac{\cos(2nt)}{4 n^2 - 1}| \le ...
1
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1answer
36 views

Product Of Series With Increment Powers

I found this interesting aptitude question and I don't know how to solve this genre of question. Any help is welcome :) $$\prod_{n=1}^{49}n^n=1¹\cdot 2²\cdot\ldots\cdot49^{49}=?$$ Thanks.
1
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4answers
26 views

Function as a series :

Let $f(x)=\sum_{n=0}^{+\infty}\dfrac{x^n}{n!}$. Verify that $$\int_0^xf(x)dt=f(x)-1$$ This is the exercise 3 of the section $7.4$, of Guidorizzi's Calculus, Vol. 4. What I have tried: By the ratio ...
0
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1answer
15 views

Solve the differential equation using power series.

$\displaystyle y^{'} = {\frac{y}{x}} + 1$ cannot be solved for $y$ as a power series $x$. Solve this equation for $y$ as a power series in powers for $ x-1 $.: Introduce $t=x-1$ as a new independent ...
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0answers
23 views

Solve this equation for y as a power series in powers of x - 1.

$\displaystyle y^{'} = {\frac{y}{x}} + 1$ cannot be solved for $y$ as a power series $x$. Solve this equation for $y$ as a power series in powers for $ x-1 $.: Introduce $t=x-1$ as a new independent ...
0
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1answer
46 views

Find the radius of convergence of the Power series $1+z+(z^2)/(2^2)+(z^3)/(3!)+(z^4)/(2^4)+…$

Find the radius of convergence of the Power series $$1+z+\frac{z^2}{2^2}+\frac{z^3}{3!}+\frac{z^4}{2^4}+\frac{z^5}{5!}\cdots $$ Put the series in the form ...
1
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1answer
57 views

Why do both trig functions have the same Macluarin series?

Both the degree version and the radian version of the trig functions have the same Maclaurin series, yet they are different. How is this possible? How can two different functions have the same ...
1
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1answer
39 views

$f_1 \in L^1_{loc}(\mathbb{R})$ and $f_{n+1} (x)= \int_0^x f_n(t) dt$, What is $\sum_n f_n$?

$f_1 \in L^1_{loc}(\mathbb{R})$ and $f_{n+1}(x) = \int_0^x f_n(t) dt$, What is $\sum_n f_n$? (and converges in what sense?) My attempt: Suppose $f_1$ is bounded, define the continuous linear ...
2
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3answers
53 views

Sum of Harmonic Numbers

Similar to this question , let $H_n$ be the $n^{th}$ harmonic number, $$ H_n = \sum_{i=1}^{n} \frac{1}{i}$$ Is there a similar method to calculate the following?: $$\sum_{i=1}^{n}iH_i$$
8
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1answer
387 views

Sum of Squares of Harmonic Numbers

Let $H_n$ be the $n^{th}$ harmonic number, $$ H_n = \sum_{i=1}^{n} \frac{1}{i} $$ Question: Calculate the following $$\sum_{j=1}^{n} H_j^2.$$ I have attempted a generating function approach but ...
2
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0answers
26 views

Need some help with this Cardinality/sets question.

I've got this problem about sets, and cardinality. I don't really understand it other than cardinality is the number of elements within each set, I don't understand a lot of the signs used within the ...
0
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1answer
21 views

Laurent series of $\frac {sin(z)}{z^2(z-\pi)}$

I have to find the Laurent series, around $z=0$ of $$\frac {sin(z)}{z^2(z-\pi)}$$ when $|z|>\pi$. I've already calculated the series separately, but I'm having trouble putting the whole thing ...
0
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1answer
18 views

Lebesgue Measure of the set of roots of a complex exponential equation

In the following equation $\{\beta_i\}_{i=1}^N$ and $\{\alpha_i\}_{i=1}^N$ are non-zero complex numbers: $\sum_{i=1}^N \beta_i e^{\alpha_i t} = 0$. I would like to know if the Lebesgue measure of the ...
0
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1answer
12 views

Finding Convergeance sum for two power-series.

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
0
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1answer
16 views

Finding the coefficients of the series solution of an initial value problem

I shell present 2 questions I came across today related to this subject. I need some explanation about the meaning of the IVP I am given. We look for a solution of the form $y=\sum a_nx^n$. So after ...
2
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0answers
22 views

Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
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3answers
40 views

Series solution to a 2nd order linear ODE

Consider the ODE $$(1+x^2)y''+3xy'+y=0$$ Find a solution in the form of $y=\sum a_nx^n$. So after doing the algebra using these 2: $$y'=\sum_{n=1}^\infty n a_n(x-x_0)^{n-1}$$ $$y''=\sum_{n=2}^\infty ...
1
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1answer
16 views

Compound interest problem with increasing deposits

An Investor starts with an initial investment : $A$ He earns a steady profit of 10 percent per year. But every year he adds additional amount which increases by 15 percent every year. At the end of ...
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0answers
17 views

Solving Power Series Equality

I'm trying to solve the following power series equation to get the coefficients $\{a_k\}$: $$\sum_{k=1} ^ \infty a_k x^k = \sum_{\ell = 0} ^\infty \frac{x}{\ell !} \left( \sum_{k=1} ^ \infty a_k x^k ...
0
votes
2answers
47 views

Showing that a generating function is equivalent to some fraction

I am working with generating functions and am required to prove that the generating function for the sequence $\{a_k\}$ where $a_k = (-8)^k$ for all integers $k\geq0$ is $\cfrac{1}{1+8x}$ and I have ...
0
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0answers
14 views

What series describes the feedback of a fully connected network with signal strength at each node converging to 1?

Take a fully connected network with $N$ nodes operating in lockstep. One and only one node will receive a signal from an external source at a time step $t$, but the node receiving it is random. The ...
3
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0answers
24 views

Intuitive explanation of proof of Abel's limit theorem

Assume the series $$f(x)=\sum_{n=0}^{\infty}a_n x^n$$ converges for $-r<x<r$. Abel's theorem says that if the series also converges at $x=r$ then $\lim_{x\to r-} f(x)$ exists and we have ...
0
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2answers
37 views

Abel's theorem - examples

I have got Abel's Theorem in this form: If a power series is converges at one of the ends of the partition of convergence, its sum is continuous at this point (one-sided). And I have got an example ...
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0answers
9 views

Power series uniqueness

I'm having trouble with a problem for my thesis. I have two analytical functions which to first order are different, does it mean that they are different functions? Remember they have a series ...
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0answers
27 views

A generalization of Clausen's formula

Clausen's formula, $${}_{2}F_{1}(a, b; c; x)^2 = {}_{3}F_{2}(2a, 2b, a + b; 2a + 2b, c; x), \quad c = a + b + \frac{1}{2},$$ is well known. Does anyone know if this formula has been generalized for an ...
1
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1answer
43 views

Determining a power series of $f(z)=\exp(z^{2})$

Let $f(z)=\exp(z^{2})$. Determine a power series of $f$, i.e the coefficients $a_{k}$ such that $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ for all $z\in \mathbb{C}$, where $a_{k}=f^{(k)}(0)/k!$. First ...
0
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1answer
58 views

How does one obtain the expansion of $e^{-x^2}$ in a power series?

So I know that the Power Series $y = \displaystyle\sum_{m=0}^\infty\displaystyle\frac{(-1)^m}{m!} x^{2m}$ is equivalent to $e^{-x^2}$. Could someone show me why this is?
0
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1answer
34 views

Power series and Fourier identity approximated in two or three iterations

I understand that Fourier has proven that the sum of sines and cosines can be used to describe (almost) any curve. The power series describe that the sum of polynoms can be used to describe (almost) ...
2
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3answers
55 views

What's the radius of convergence for power series of $1/(1-(x+x^2))$? Is it symmetrical?

When reading the section on composition of power series in Book Calculus With Analytic Geometry (George F. Simmons) (2nd edition, pp. 517), the author claimed you can replace $x$ in a power series ...
6
votes
1answer
66 views

Can we characterize the space of functions which is real analytic but not real entire?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
1
vote
1answer
33 views

Series Solutions Near an Ordinary Point

I am attempting to solve this problem for practice: $y"-(x-3)y' - y = 0$ at $x_{0} = 3$. But it appears as though I don't have an idea of the best approach to employ to go about solving it. Can ...
1
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1answer
33 views

Can I solve an Euler differential equation by using the Frobenius method?

I'm having some trouble by trying to solve Euler equations by using the Frobenius method. For example, I'm asked to solve the Euler differential equation $$ x^2y'' + xy' - y = 0 $$ using a power ...
0
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0answers
11 views

Series Solution of Linear Second Equations - Difficulty Formatting Final Answer

I have been working the following differential equation: \begin{align} (1-x^2)y'' -8xy' -12 y = 0 \end{align} which has solution \begin{align} y= a_{0}\sum_{m=0}^{\infty}(m+1)(2m+1)x^{2m} + ...
1
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1answer
21 views

Index of Summation Shift? Power Series and Differential Equations

I have never had to index shift a summation series before, and it seems relatively straightforward, however, I am looking at an example in my textbook that doesn't make sense. I am wondering if ...
0
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0answers
26 views

Power series with complex variables inequality

I am struggling to prove the following inequality: For $z \in \mathbb{C}, r \in \mathbb{R}, n \in \mathbb{N}$, if $|z| \leq r$ and $1 \leq r < n$ then ...
6
votes
3answers
133 views

Use Taylor Series method to solve $y''-2xy+y=0$

I am doing some practice problems for solving second order ODEs, and I am a bit stuck on this one. Here is what I have: $y''-2xy'+y=0$ Let $y = \sum_{n=0}^{\infty} C_nx^n \implies y' = ...
2
votes
1answer
47 views

Showing that if the $n$th derivative of a function is bounded then it is real analytic

I reproduce from my lecture notes: Suppose $f$ is $C^\infty$ on $[a,b]$ with $$\left|f^{(n)}(x)\right|\leqslant M~~\text{for all}~~x\in(a,b).$$ Then $f$ is real analytic in $[a,b]$. Proof. ...
0
votes
1answer
13 views

Showing that two infinite series converge to the same value

I was preparing for my exam and came across this problem. Show that The series on the left hand side is the power series of $\ln(1+x)$ evaluated at $x=1$. This is what i've done so far. From ...
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2answers
104 views

Proof that $\dfrac{1}{e^x}=e^{-x}$ without converting it to $e^{x}e^{-x}=1$.

I want to show that $\dfrac{1}{e^x} = e^{-x}$ from the Taylor expansion of $e^x$. To express $\dfrac{1}{e^x}$ as a power series, I let: $$ \left(\dfrac{1}{0!}x^0 + \dfrac{1}{1!}x^1 + ...
1
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1answer
62 views

Showing that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$ for formal power series

I've just come across formal power series and am not very fluent with them yet. I'd like to show that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$. Can anybody help?
2
votes
1answer
78 views

Summation of exponential series [duplicate]

Evaluate the limit: $$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
1
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1answer
36 views

Radius of convergence of power series $\sum_{n=1}^{\infty}{\frac{\sin n!}{n!}} {x^n}$

The power series $\displaystyle\sum_{n=1}^{\infty}{\frac{\sin n!}{n!}} {x^n}$ has radius of convergence $R$, then $R\geq1$ $R\geq e$ $R\geq2e$ All are correct I wanted to know how will get the ...
13
votes
2answers
329 views

Ramanujan's approximation for $\pi$

In 1910, Srinivasa Ramanujan found several rapidly converging infinite series of $\pi$, such as $$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. ...
3
votes
1answer
58 views

Showing $y_1$ or $y_2$ are not polynomials

proof that $y_1$ or $y_2$ are not a polynomial for any $n$ $$ y_1(x)=1-\frac{n(n+1)}{2!}x^2+\frac{(n-2)n(n+1)(n+3)}{4!}x^4-+\cdots$$ $$ ...
4
votes
1answer
80 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
1
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1answer
83 views

Product of two $2$-variables Taylor series

Using the standard multi-index notation, suppose we have the two Taylor series $$ f(\theta) := \sum_{|\alpha|=0}^{\infty} a_{\alpha} \theta^{\alpha} $$ and $$ g(\theta) := \sum_{|\alpha|=0}^{\infty} ...
3
votes
0answers
166 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
1
vote
1answer
27 views

Radius of convergence of powerseries containing $(\log n)^n$

$$ \begin{align} \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \end{align} $$ What is the radius of convergence of this power-series? I tried applying the root test and the ratio test , but I couldn't ...
0
votes
2answers
35 views

Power series solution to integral equation

Hi guys i'm reading a paper in which the authors have two coupled integral equation for the function $f(x)$ and $g(x)$, in order to solve this problem they employ a power series expansion of these ...
0
votes
2answers
47 views

Disk of convergence of the series $ \sum\limits_{n=1}^\infty n!\,(z-i)^{n!} $

$$ \sum_{n=1}^\infty n!(z-i)^{n!} $$ Find the disk of convergence of this powerseries. Can I set $n!=k$ and then deal with $\sum_{n=1}^\infty k z^k$ . On another note $\frac{z^{(n+1)!}}{z^{n!}}$ ...