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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2answers
40 views

Definite Integral, use a power series.

Use a power series to approximate the definite integral, I, to six decimal places. I tried to definite integral, but the answer is wrong. Where did I make a mistake? $$\int_{0}^{0.2} \ln(1+x^4)\ dx ...
2
votes
0answers
39 views

Writing $\frac{1}{1 + w + w^2}$ as a power series and finding the ROC

I have to write the following: $\frac{1}{1 + w + w^2}$ as a power series: $$\sum_{n=0}^{\infty}{a_nw^n}$$ and find the radius of convergence of the series (in the complex plane). Obviously you can ...
3
votes
1answer
70 views

Show that each of the coefficients of a complex power series are real

Suppose that $g(z) = \sum c_nz^n$ has radius of convergence $R > 0$ and that $g(\frac{1}{m})$ is real for all $m > \frac{1}{R}$. Show that each $c_n$ is real. I know that if $R>0$ ...
1
vote
1answer
38 views

Determining a complex function represented as a series

Find the domain $D$ of the function $ f(z) = \sum (-1)^n(z-1)^n - \sum i^{n-1} (z-i)^n $ Then determine $f(z)$, for all $z \in D$ I figured for the first part of the function i could ...
1
vote
0answers
73 views

Asymptotic Expansion for a Function involving a Weird Integral

So I'm trying to find the asymptotic expansion as $x \to \infty$ of $$f(x)=\frac{1}{\bigg[A-\int \frac{\lambda^x}{\Gamma(x+1)}dx\bigg]^\frac{1}{\alpha}}$$ Note that $\lambda>0$ and $\alpha>0$. ...
1
vote
2answers
70 views

Series expansion of $ \frac{x}{\ln (1+x)}$ [closed]

What are coefficients in the expansion of series for $$ \frac{x}{\ln (1+x)} = \sum_{n=0}^\infty A_n \frac{x^n}{n!}?$$ Do they have a name?
1
vote
1answer
52 views

Does the Abel sum 1 - 1 + 1 - 1 + … = 1/2 imply $\eta(0)=1/2$?

If $\sum_{n=1}^\infty a_n$ is Abel summable to $A$, then necessarily $\sum_{n=1}^\infty a_n n^{-s}$ has a finite abscissa of convergence and can be analytically continued to a function $F(s)$ on a ...
4
votes
1answer
72 views

Proving complex trigonometric identity using power series

Prove $2$cos$^2(z) = 1+$cos$(2z)$ using power series. I know that cos$(z) = \sum (-1)^n\frac{z^{2n}}{(2n)!}$ I also know that if $a(z) = \sum a_nz^n$ and $b(z) = \sum b_nz^n$ then $a(z)b(z) ...
0
votes
1answer
15 views

Find (with proof) the domain of each of the following power series functions

Four functions here I am not quite sure how to go about doing this. I feel I am missing something very easy. Can someone please help? I am preparing for an exam and this is important.
1
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0answers
33 views

Power Series Solution to ODE

Trying to find the general solution of the following equation: $$2x(1+2x^2)y''-y'-24xy=0 $$ using: $$y(x)=\sum_{n=0}^\infty a_nx^{n+r}$$ r and $a_n$ are constants I get so far as the recurrence ...
3
votes
3answers
109 views

Calculate $\sum\limits_{i=2}^\infty\sum\limits_{j=2}^\infty \frac{1}{j^i}$

Calculate $\sum\limits_{i=2}^\infty\sum\limits_{j=2}^\infty \frac{1}{j^i}$. I am trying to figure out how to calculate this. I know it must be ...
1
vote
1answer
45 views

Formula for a geometric series weighted by binomial coefficients (sum over the upper index):$\sum_{i=0}^L {n+i\choose n}\ x^i =\ ?$

The binomial sum is $$\sum\limits_{i=0}^n {n\choose i}\ x^i = (1+x)^n,$$ where $\displaystyle{n\choose i}=\frac{n!}{(n-i)!i!}.$ Is there a corresponding formula when you sum over the upper index of ...
1
vote
1answer
32 views

Want to check that $\sum_{j=0}^{k-1}w^{ jm}=0$, $m\not\equiv 0 \pmod{k}$ where $w=e^{2\pi i/k}$

If $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$, then $$ \sum_{n=0}^{\infty}a_{kn+m}x^{kn+m}=\frac{1}{k}\sum_{j=0}^{k-1}w^{-jm}f(w^j x) \tag{1},$$ where $w=e^{2\pi i/k}$ is a primitive $k$th root of ...
0
votes
0answers
44 views

How to find the Laurent series for $\frac{1}{(z-1)(z-2)}$?

How to find the Laurent series for $$\frac{1}{(z-1)(z-2)}$$ on $|z|<2$ Here is what I did so far: $$\frac{1}{(z-1)(z-2)}=\frac{1}{(z-2)}-\frac{1}{(z-1)}$$ now ...
3
votes
1answer
44 views

Why does $\frac{1}{1-z}=-\sum_{n=0}^{\infty} \frac{1}{z^n}$ if $|z|>1$

Why does $$\frac{1}{1-z}=-\sum_{n=0}^{\infty} \frac{1}{z^n}$$ if $|z|>1$ I know the case for $|z|<1$ and if $|z|>1 \implies |1/z|<1$ so $$\frac{1}{1-\frac{1}{z}}=\sum_{n=0}^{\infty} ...
3
votes
1answer
48 views

Closed form for $\prod_{n=0}^\infty (1-z^{2^n})$

Is there a closed form for the product $$f(z) = \prod_{n=0}^\infty 1-z^{2^n}$$ either as a formal power series or as an analytic function in the disk $|z| < 1$? It's not hard to see that Taylor ...
1
vote
3answers
56 views

Finding the value of $\sum_{k=1}^{\infty} \frac{x^{4k-3}}{4k-3}$

This is my power series: $$\sum_{k=1}^{\infty} \frac{x^{4k-3}}{4k-3}$$ I need to find the sum of it. Unfortunately, I have kind of no idea how to do it. I think I need to substitute with ...
0
votes
0answers
21 views

A linear operator that extends the summation of Dirichlet series

Consider a vector space $\mathcal{V}$, a linear operator $L$ and a vector subspace $\mathcal{A}$ such that for all $x\in\mathcal{A}$ $Lx\in\mathcal{A}$ and for a number $R\neq0,1$ $R^{-1}$ is not a ...
1
vote
1answer
13 views

Prove that if there exists $c,M>0$ such that $|a_nc^n|\leq M\forall n$, then $(-c,c)\subseteq\operatorname{Conv}(\sum a_nx^n)$

Prove that if there exists $c,M>0$ such that $|a_nc^n|\leq M\,\forall n$, then $(-c,c)\subseteq \operatorname{Conv}(\sum a_nx^n)$. I tried to use the ratio test to, but I only get that $c\limsup ...
1
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0answers
19 views

Properties of Expected Value of Hermite Polynomial

Are there well know properties of \begin{align} E[ H_n(V)] \end{align} where $H_n( \cdot)$ is a Hermite polynomial and $V \in L^n$? For example if $V$ is Gaussian with variance $1$ and mean $\mu$ ...
6
votes
0answers
33 views

Certain symmetrized product of cosines - can it be transformed into more manageable form

I am interested in the following expression: $$ F_{k_1,\ldots,k_n}(t):=\sum_{\sigma\in S_n}\cos(\sigma(1)k_1t)\cos(\sigma(2)k_2t)\cdots\cos(\sigma(n)k_nt) $$ where $k_1, \ldots, k_n$ are natural ...
1
vote
2answers
34 views

Interval of convergence of $\sum_{n=1}^{\infty} \frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)x^n}{n!}$

Given the series $$ \sum_{n=1}^{\infty} \frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)x^n}{n!} \quad \quad k \geq 1 $$ Find the interval of convergence. I started by applying the Ratio ...
2
votes
0answers
68 views

Power Series Solution to $(\cos x)y'' + xy' - 2y = 0$?

I got the following question for some homework, and I am struggling with it. I know how to show they form a fundamental set of solutions and I have a guess at the radius of convergence. I am just ...
0
votes
1answer
50 views

What's the function that is related to 3 as the Riemann zeta function is related to 2?

For $f(x)=\sum_{n=0}^\infty a_n x^n$, a real number $R\neq 1$, $g(x)=f(x)-Rf(x^2)$ Abel summable at $x=1$, $g(1)=\lim_{x\to 1^-} g(x)$, the elementary Ramanujan sum of $f(x)$ at $x=1$ is defined by ...
1
vote
3answers
93 views

Calculate $a_n = \binom{n}{2} + \binom{2}{n}$

Calculate $a_n = \binom{n}{2} + \binom{2}{n}$ Could you give me a hint how to start solving this equation? How can I expand $\binom{2}{n}$? Definition of $\binom{a}{b}=\frac{a \cdot (a-1) \cdots ...
0
votes
1answer
33 views

Calculate coefficients of power series

Calculate the coefficients of the power series expansion of $f(z)=\frac{2}{\sqrt{1-3z}}+\frac{1}{(1-z)(1-2z)}$ Could you check if I understood the task and calculated it correctly? ...
1
vote
3answers
65 views

Taylor Series of $f(x) = \sqrt{x}$ about $c = 1$ [duplicate]

Taylor Series of $f(x) = \sqrt{x}$ about $c = 1$ I've tried doing this problem but stuck at finding a pattern.. Work: $$T_n = \sum^\infty_{n=0}\frac{f^n(c)}{n!}(x-c)^n = f(a) + ...
2
votes
1answer
57 views

Find the Laurent series for $f(z)=\frac{2}{(z-4)}-\frac{3}{(z+1)}$

The question is 2 parts - I'm to find the Laurent series valid for $$1 < |z| <4$$ and $$|z| > 5$$ I've already solved the first part, but I have a conceptual question about the second part. ...
0
votes
2answers
74 views

Power series radially vanishing or exploding on their circle of convergence

I'm trying to understand the behavior of two kinds of power series on the complex plane. Let's call them $f(z) = \sum_{n \in \mathbb{N}} a_n z^n$, and suppose the radius of convergence is $1$, so ...
0
votes
1answer
33 views

Can an alternating series be thought of as a geometric series? [closed]

Can $\sum(-1)^{n}$ be thought of as a geometric series? Although we know that $\sum(-1)^{n}$ diverges by the definition of Alternating Series. Can we say that it also diverges by using Geometric ...
1
vote
2answers
44 views

Intersection of a power function with a line: how to compute?

How to compute $x$ from $$q x^p = 1 - x$$ where $x$ and $q$ are positive, while $p$ is a real number? When $p > 0$: it's two monotonic functions, one increasing and one decreasing, and having ...
2
votes
4answers
69 views

Finding the polynomials (or power series) $p_n(x)$ such that $2\int x^ne^{x^2}dx = p_n(x) e^{x^2}$

By using integration by parts, one can obtain the following equation $$2\int x^ne^{x^2}dx=x^{n-1}e^{x^2}-(n-1)\int x^{n-2}e^{x^2}dx .$$ A recursive formula for $\int x^ne^{x^2}dx$ that works if ...
1
vote
1answer
43 views

Derivative of quotient of a formal power series

Suppose $G(x)$ is a formal power series with non-zero constant term. How could I show that $$\Big(\frac{1}{G(x)}\Big)'= -\frac{G'(X)}{G^2(X)}$$
1
vote
2answers
47 views

Formal power series (multiplication, divison)

Let be $F$ the generator function of the prime numbers and $G(x) = \sum_{k=0}^{\infty}(-2)^{k}x^{k}$, so $$F=2+3x+5x^2+7x^3+ \dots \\ G=1-2x+4x^2-8x^3+ \dots$$ a) What will be the first four ...
0
votes
1answer
29 views

How do you find the incidental equation when using the Frobenius method?

If I'm not mistaken, the Frobenius Series is given by $$ y = a_{0}x^{r} + a_{1}x^{r+1} + a_{2}x^{r+2} + \dots + a_{n}x^{r+n} + \dots $$ and so we also have $$ y' = ra_{0}x^{r-1} + (r+1)a_{1}x^{r} + ...
0
votes
1answer
43 views

Series Solutions for Differential Equation (similar to Airy function)

I have to find a series solution to: $$y'' - t^3 y =0$$ I go through the steps like so, $$\sum_2 n(n-1)a_n x^{n-2} - \sum_0 a_n x^{n+3} = $$ $$\sum_0 (k+2)(k+1)a_{k+2} x^{k} - \sum_3 a_{k-3} x^{k} ...
1
vote
1answer
21 views

Find the power series of the 3 functions together

$$f(z)=3e^z+2ze^z+z^2e^z$$ In the first part of the problem, I had to find the power series of each term, which are listed below. $$3e^z=\sum_{n=0} \frac{3z^n}{n!}$$ $$2ze^z= \sum_{n=0} ...
4
votes
0answers
71 views

Asymptotic Expansion for an Integral

So I have $$\psi(x)=\int_{2\lambda}^x \frac{(e\lambda)^{z}}{z^{z+1/2}} dz$$ I'm trying to find the asymptotic expansion of $\psi(x)$ as $x \to \infty$ for as many orders as possible. How would I go ...
2
votes
1answer
27 views

If $f,g$ real analytic and $\lim_{t \to t_0} f(t)/g(t)$ exists then $f/g$ is analytic

If $f,g$ are real analytic at $t_0$ and $\lim_{t \to t_0} f(t)/g(t)$ exists then is it true that $f/g$ with the limiting value filled in at $t= t_0$ is real analytic at $t_0$? I know the complex ...
0
votes
1answer
17 views

how to find out radius of convergence of the following series?

let $f(x)=\frac{1}{1+x^2}$. Consider its taylor series expansion about a point $a$ in $\mathbb R$ given by $f(x)=\sum_{n=0}^\infty a_n(x-a)^n$. what is the radius of convergence of this series? one ...
0
votes
2answers
48 views

Calculating $\sum_{n \geq 0} n^2 x^n$

My attempt so far: $$\sum_{n \geq 0} n^2 x^n = x \sum_{n \geq 0} n^2 x^{n-1} = x \sum_{n \geq 0} n (x^n)'$$ And now I've got stuck. How can I continue from now?
1
vote
1answer
38 views

Growth estimate of an entire function

I have not even understood the statement clearly to attempt it ! Suppose that $f$ is an entire function and that there exist two real numbers $M > 0$ and $p ≥ 1$ such that $|f (z)| ≤ M (1 + |z|^p ...
4
votes
1answer
40 views

IEEE 754-like definition of “real” real numbers

once I have mapped, geometrically or by $\sin\left( \arctan\left( x \right) \right)$ the range $\left[ 0,+\infty \right[$ into $\left[ 0,1 \right[$, it's very "nice" to write (real) numbers in ...
1
vote
0answers
30 views

Performing Taylor series with 'fractional' powers

Taylor expansions are used all the time for physics problems, but sometimes they don't work because we expanded in the 'wrong' parameter. For example, suppose we're doing a relativistic kinematics ...
1
vote
1answer
31 views

Calculating sum

$$a_n=n+n \cdot 5^n \quad n \geq 0$$ $$b_n=\sum_{k=0}^{n-1} a_k \quad n \geq 1, b_0=0$$ Find explicit expression for: $$\sum_{n \geq 0} b_n x^n$$ So we have $\sum_{n \geq 0} \Big( ...
0
votes
2answers
63 views

Taylor series expansion of $ f(x)=e^{-x^2}$

How to find Taylor series expansion of $f(x)=e^{-x^2}$ What I'm stuck at is proving that the error $$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$$ of the expansion tends to zero.
0
votes
2answers
61 views

Square Root of Formal Power Series [closed]

Suppose $R(x)$ is a formal power series. If $[R(x)]^2=x+1$, must be $R(x)$ be the formal power series of $\sqrt{x+1}$, or $-\sqrt{x+1}$?
1
vote
1answer
32 views

Proving $|R(z_1)-R(z_2)|\leq |z_1-z_2|$ where $R(z_1)$ is the radius of convergence of $f$ around $z_1$.

Trying to prove this: Let $D$ be a disk and $(f,D)$ a function element. If $R(z_1)$ is the radius of convergence of the power series expansion of $f(z)$ about a point $z_1\in D$ then ...
0
votes
1answer
51 views

Is the linear subspace of elementary Ramanujan summable series closed under iteration?

For $f(x)=\sum_{n=0}^\infty a_nx^n$ and $g(x)=f(x)-Rf(x^2)$, $R\neq 1$, the series $\sum_{n=0}^\infty a_n$ belongs to the elementary Ramanujan class $R$ if $g(x)$ converges at least for $|x|<1$ and ...
1
vote
2answers
29 views

The number of solutions for infinite power series equations $\sum_{k=0}^{\infty}A_k z^k=0$?

For polynomial equations we have the fundamental theorem of algebra, but what about infinite power series equations? For example, by using Taylor series for different functions we can have: ...