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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0
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0answers
35 views

First develop the function $\sqrt{x}$ in a series of powers of $(x-1)$ and then use it to approximate $\sqrt{0.9999999995}$ to ten decimal places. [on hold]

First develop the function $\sqrt{x}$ in a series of powers of $(x-1)$ and then use it to approximate $\sqrt{0.9999999995}$ to ten decimal places. I'm stuck on how to do this problem. Any solutions ...
2
votes
0answers
16 views

Fractional Euler sums?

As we know, the classical linear double Euler sums is defined by $${S_{p,q}} = \sum\limits_{n = 1}^\infty {\frac{{{\zeta _n}\left( p \right)}}{{{n^q}}}} \;$$ where $p, q\ (q \ge 2)$ are positive ...
2
votes
2answers
36 views

What function is this? $\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$ [on hold]

I am interested to know what function represents the following series: $$\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$$
4
votes
2answers
60 views

If $\sum_{m,n}a_{mn}x^m(1-x)^n\equiv 0$, can we conclude $a_{mn}=0$?

Assume $\{a_{mn}\}$ are some real numbers between -1 and 1. If we know $$\sum_{m,n}a_{mn}x^m(1-x)^n\equiv0\quad\forall x\in(0,1),$$ can we conclude that $a_{mn}=0$ for all $m,n\geq 0?$ Thanks.
-5
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6answers
58 views

What is $1+2+4+8+16+…+2^n$? [duplicate]

What is the result of: summation from one, two, four, eight until $n$ power of two? Thank you!
0
votes
0answers
29 views

integral of complex function, power series

let $\mu$ be a finite borel measure on $[0,+\infty)$ and let $f$ be defined by $$f(z)=\int_{[0,+\infty)}\frac{d\mu(t)}{t-z},\quad z \in \mathbb{C} \setminus [0,+\infty)\,.$$ *show that the integral ...
0
votes
1answer
31 views

Series expansion for innocent looking function

$$f(z) = \frac{1}{z^2}$$ is given, where $f(z)$ is complex valued function. How can one find series expansion at $ z=i$ with using geometric series approach? It seems simple but first tries gives ...
1
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3answers
42 views

Evaluating the ratio $ {{a_{n+1}}\over{a_n}}$ in calculating the radius of convergence for a power series

In calculating the radius of convergence for the power series $$ \sum_{n=1}^\infty {{(2n)!}\over(n!)^2}\ x^n $$ By the ratio test, we let $$ a_n = \lvert {{(2n)!}\over(n!)^2}\ x^n \rvert \quad\quad ...
1
vote
2answers
40 views

About complex power series

I have a really big doubt. I'm trying to find all the values of $z$ for which the next power series converges: $$\sum_{n=0}^{\infty} \frac{z^{3n}}{8^{n}(1-in)} $$ Using the root test I have that ...
2
votes
0answers
47 views

Positivity of an alternating series.

Greetings esteemed mathematicians. I've managed to prove that the following series \begin{equation} f_{\lambda}(\omega)= ...
1
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0answers
33 views

Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a ...
3
votes
1answer
74 views

Finding a power series representation for $\left(\frac{x}{2-x}\right)^3$

Find a power series representation for $\displaystyle\left(\frac{x}{2-x}\right)^3$ My approach is in finding something similar to $\displaystyle\left(\frac{x}{2-x}\right)^3$ to which I can easily ...
1
vote
3answers
183 views

How to calculate the limit of this sum with different methods? [duplicate]

It's a basic question , but what are the common methods to calculate limits like this one: $$\sum_{k=1}^\infty \frac{3k}{7^{k-1}}$$
0
votes
1answer
59 views

How to calculate this sum?

Let $x_1,\cdots,x_k$ be numbers between 0 and 1. Then is it possible to get explicit expression for the following sum:$$\sum_{n_1,\cdots,n_k\geq 1} x_1^{n_1}\times C_{n_1+n_2}^{n_2}\times ...
1
vote
1answer
32 views

Power series of z/sin(z)?

So I need to compute the coefficient of the $z^4$ in the power series of $\frac{z}{\sin z}$. I tried differentiating the function and obtaining coefficients like in Taylor's expansions but had a ...
14
votes
3answers
172 views

Show that $\frac{x}{3!}-\frac{x^3}{5!}+\frac{x^5}{7!}-\cdots\leq \frac{1}{\pi}$.

My problem is to show that $$\frac{x}{3!}-\frac{x^3}{5!}+\frac{x^5}{7!}-\cdots\leq \frac{1}{\pi}$$ for all $x\in\Bbb R$. I was thinking of first finding the max and then show that its less ...
1
vote
1answer
47 views

Multiplicative inverse of the power series $e^x - c$ for $c \neq 1$.

We know that the power series $f(x)= e^x -c \in \mathbb C[[x]]$ for $c \neq 1$, has a multiplicative inverse, since it's constant coefficient is non-zero. I was wondering whether the inverse is known ...
0
votes
3answers
50 views

Why radius of power series is defined as $\lim_{n \to \infty} (a_n)^{1/n} = 1/R$

I am reading definition of radius of convergence of power series $a_nx^{n}$ as $\limsup_{n \to \infty} (|a_n|)^{1/n} = 1/R$. I cannot understand it intutively, it makes no sense to me. Can anyone ...
0
votes
1answer
28 views

Finding a power series representation for $\frac{1}{(2-x)^2}$ in powers of $x$

Problem: Determine a power series representation for the function \begin{align*} \frac{1}{(2-x)^2} \end{align*} in powers of $x$. On what interval is the representation valid? Attempt: We have ...
1
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0answers
56 views

We have $ f(x) = \sum_{n \geq 1} \frac{(x-1)^n}{n}$ prove that $f(x) = -\ln(2-x)$.

I am having problems with the following exercise, I have solved the first two parts of the exercise but I am unsure about the last part. I have the following power series $$f(x) = \sum_{n \geq 1} ...
3
votes
2answers
55 views

Are there many different power series representation for a given function?

So I have to find the power series representation for $f(x) = \ln (3-x)$. I attempted the following: $$\ln(3-x) = \int {- \frac{1}{3-x} dx}$$ $$ = - \int { \frac{1}{1-(x-2)} dx}$$ $$ = - \int ...
-1
votes
3answers
39 views

Differentiate the following power series $\sum_{n \geq 1} \dfrac{(2x-2)^n}{n2^n+1}$

I am having issues with the differentiation of the following power series $$ \large f(x) = \sum_{n \geq 1} \dfrac{(2x-2)^n}{n2^n+1}$$ I get the following result $$ \large f'(x) = \sum_{n \geq 1} ...
1
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2answers
51 views

Find $\sum_{n=1}^{\infty}a_nx^n$ given $a_0=3, \ 3na_n+3(n-1)a_{n-1}=2a_{n-1}$

Given $\ a_0=3$, $\,3na_n+3(n-1)a_{n-1}=2a_{n-1}$, find $\ f = \sum_{n=1}^{\infty}a_nx^n$. I have proved that when $\ \left\lvert x \right\rvert<1$, this exponential series function is convergent. ...
0
votes
1answer
78 views

Taking Limits of Sets

I know this sounds like a ridiculous idea- but it's the only one I can think of for this radius of convergence problem for a power series involving sine. I want to let $P:= \{k:|sin(k)| \geq \delta ...
0
votes
0answers
42 views

Summation of an infinite series

For $0<\theta\leq 1$ and $A, B \geq 1$, we wish to find summation (or upper bound) of the following infinite series: $$ \frac{(\theta)^2 }{A^{(\theta^{\frac{1}{2}})}B^{(\theta^{\frac{1}{2}})}} ...
2
votes
1answer
63 views

Power series expansion of an Operator.

I've been reading a paper called "Separation of variables for the quantum $Sl(2,R)$ spin chain" in which the author at one point does a power series expansion I do not understand. The problem is this ...
0
votes
2answers
19 views

Generalized Matrix Series

I have to sum a matrix series of the form: $$ \sum_{s=0}^\infty M^s B R^s $$ Is it possible to obtain a closed form formula as in the usual geometric series? Thanks.
0
votes
1answer
32 views

coefficients of a power series

I have the function $$f(x)=\frac{2x}{10+x}$$ and I am asked to find its power series representation which I found to be $$\sum_{n=0}^{\infty} (-1)^{n} *\frac{2x^{n+1}}{10^{n+1}}$$ and I found the ...
7
votes
2answers
215 views

Sum of 1.5-powers of natural numbers

I recently have met the following approximate equation: $$\sum_{k=1}^n k^{1.5}\approx\frac{n^{2.5}+(n+1)^{2.5}}{5}.$$ It's a rather accurate approximation (for $n=40$ the absolute error is $\approx ...
0
votes
1answer
32 views

Finding series representation of $\frac{1}{P(D)}$ through ordinary division

I am studying ODEs from ordinary differential equations by Tenenbaum and Pollard. The book in its fifth chapter explains inverse operators for finding the particular solution of a constant coefficient ...
1
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3answers
50 views

Being careful with terms of infinite sums

$ cos(x): = \sum_{k=0}^\infty \frac{(-1)^nx^{2n}}{2n!}$ $=1- \frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}...$ I would like to show that for $x \in [0,2]$ $cos(x) \leq 1- ...
7
votes
5answers
157 views

How to show that $e^{-x}$ tends to $0$ when $x\to \infty$ if $e^{-x}$ is defined as the power series.

With only the formal definition of $$f(x) = \exp(-x)= \sum \frac{(-x)^n}{n!}$$ how can we show that $$\lim_{x\to \infty} f(x)=0?$$ I am looking for a proof that would not use the identity ...
0
votes
1answer
40 views

An upper-bound problem of sum of positive numbers

I came across the following problem of inequality. If $ \ \ \sum_{i=1}^{n}x_i^3\leq S$ then find the value of $K$ such that $\sum_{i=1}^{n}x_i\leq K$. It is given that $x_i>0,\forall i\in ...
0
votes
1answer
31 views

Solve for power series satisfying certian relations among each other

Write $y_i = f_i(z) = \sum_{k=1}^{\infty} c_k^{(i)} z^k$ for $i=1,\ldots, i$ for four (formal) power series. The following relations are given between them \begin{align*} y_1 & = y_2 + y_3 \\ ...
1
vote
1answer
75 views

Can you define the radius of convergence of a power series by an upper bound on the sequence of coefficients?

Let $P(z) = \sum_{n = 0}^\infty c_n z^n$ be a complex power series. Consider the follwing subsets of $\mathbb{R}$ $$ \begin{align} A_1 &:= \{r \geq 0 \,:\, (c_n r^n)_{n \in \mathbb{N}_0} \text{ ...
0
votes
1answer
39 views

Are these two power series equal?

Let $f(x)=\sum_{n=0}^\infty a_n x^n$ and $g(x)=\sum_{n=0}^\infty b_n x^n$ where $a_n,b_n\in[0,1]$ for all $n\geq 0$. Hence we know these two power series are convergent on $(-1,1)$. Now assume there ...
3
votes
2answers
190 views

Power series expression for $\exp(-\Delta)$

I know it should be true, but for some reason I can't get the calculations to work out in order to show that if $f$ is smooth and compactly supported, the power series $\sum_{j=0}^\infty ...
1
vote
0answers
105 views

How did Euler give a sum to the divergent series $…x^{-3}+x^{-2}+x^{-1}+1+x^1+x^2+x^3.. = 0$?

In Prof Norman Wildberger's A Socratic look at the logical weaknesses of modern pure mathematics (which just made available on youtube), he mentioned a discovery by Euler (30:55) that: ...
9
votes
4answers
126 views

Prove the series has positive integer coefficients

How can I show that the Maclaurin series for $$ \mu(x) = (x^4+12x^3+14x^2-12x+1)^{-1/4} \\ = 1+3\,x+19\,{x}^{2}+147\,{x}^{3}+1251\,{x}^{4}+11193\,{x}^{5}+103279\, ...
1
vote
2answers
62 views

series function

We know that there are some series that can be written in short, for example: $$ \sum_{n=0}^\infty x^n=\frac{1}{1-x},\qquad |x|<1 $$ Is there similar function for $$ \sum_{n=1}^N x^{1/n} $$ or $$ ...
1
vote
0answers
22 views

Simple example of approximating a nonlinear system with Volterra series

I'm trying to understand Volterra series as a means of modelling/approximating nonlinear input-output relations. I'm having trouble to understand the abstract definitions of kernels/functionals and ...
0
votes
2answers
57 views

Differential equation $f'''(x)=-f(x)$ with restriction using power series

Using power series, Prove the existence of a $C^3$ function (continuously differentiable 3 times) $f:\mathbb{R} \to \mathbb{R}$ such that $f'''(x)=-f(x)$ $\forall x \in \mathbb{R}$ and ...
0
votes
3answers
91 views

Calculate the sum of this series

$$ \sum_{n=1}^\infty \frac{1}{n^2 3^n} $$ I tried to use the regular way to calculate the sum of a power series $(x=1/3)$ to solve it but in the end I get to an integral I can't calculate. Thanks
6
votes
2answers
80 views

Derivative of sum of powers

For fixed $n \geq 1$ and $p \in [0,1]$, is there a nice expression for the derivative of $\sum_{k=0}^n p^k (1-p)^{n-k}$ with respect to p?
4
votes
1answer
37 views

On finding special kinds of power series

Let $\sum a_n x^n$ be a real power series with finite positive radius of convergence $R$, then is it true that for every real number $s>0$ , we can find a real sequence $\{b_n\}$ (depending on $s$, ...
2
votes
3answers
73 views

Prove that series $ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $ and $ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $ have the same radius of convergence.

I want to prove that these two power series $$ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $$ and $$ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $$ have the same radius of convergence. What I've done so far is: ...
0
votes
1answer
23 views

Summation of finite power seires

Is it possible to find a close form solution for $S_1$. $S_1$ is defined as follows: $S_1=\sum_{k=b}^{\infty}\frac{x^k}{k!}$ ; Where $0<x<b<\infty$ If $b=0$ then $S_2 = e^x$. But how do we ...
11
votes
2answers
357 views

Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
0
votes
1answer
48 views

Extract $A+B+C$ from $A^{\frac{3}{2}}+B^{\frac{3}{2}}+C^{\frac{3}{2}}= R*D^{\frac{3}{2}}$

I need to find $A+B+C=?$ from $A^{\frac{3}{2}}+B^{\frac{3}{2}}+C^{\frac{3}{2}}= R*D^{\frac{3}{2}}$ I know that I can't use log for this equation. Do anyone have any ideas of how to do the ...
1
vote
6answers
81 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...