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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24 views

The range of validity for the sums of power series

If I have a power series:($z$ is complex here) $\displaystyle \sum_{n=0}^{\infty}z^n$ valid for |$z|<1$ and another $\displaystyle \sum_{n=0}^{\infty}({\frac{z}{2}})^n$ valid for |$z|<2$ I ...
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1answer
24 views

Expressing $z\in\mathbb{C}[[w]]$ as a power series in $y\in\mathbb{C}[[z]]$.

I'm given that $$w=z+\sum_{i=2}^\infty a_iz^i$$ $$z=w+\sum_{i=2}^\infty b_iw^i$$ $$y=z-\sum_{i=2}^\infty (-1)^ia_iz^i$$ And that those series are all convergent (in particular I'm not given that they ...
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2answers
33 views

Impossible interval for convergence of a power series. [closed]

For the series $$\sum_{n=0}^{\infty} a_n(x-2)^n$$ Which of the following intervals is impossible for the convergence of that series? $$1≤x≤3$$ $$1≤x≤4$$ $$0≤x≤4$$ $$-1≤x≤5$$
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0answers
13 views

Convergence of integrated power series.

If I integrated a series that included the endpoints in the interval of convergence, will the new series also have those endpoints in its own interval of convergence?
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1answer
46 views

Compounded Interest with Exponentially Increasing Periodic Payments

Given the formula $$v_a = p\left(\frac{\left(1+\frac{r}{n}\right) ^{nt}-1}{\frac{r}{n}}\right)$$ for the value $v_a$ of an account growing at a periodic rate $r$ with a regular deposit $p$ compounded ...
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1answer
24 views

The autocovariance function of ARMA(1,1)

So I am reading Brockwell and Davis introduction to Time Series analysis on page 89 where he derives the ACVF of an $ARMA(1,1)$ given by: $X_t - \phi X_{t-1}=Z_t+\theta Z_{t-1}$ with ${Z_t}$ is ...
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1answer
47 views

$\sum_{n=0}^\infty z^n = \prod_{m=0}^\infty \left(1+z^{2^m}\right)$

When reading Iwaniec and Kowalski's Analytic Number Theory, I came across the following "identity" on page 11 (the Amazon link has a free book preview which includes page 11): $$\sum_{n=0}^\infty z^n ...
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2answers
44 views

How to evaluate the $\lim\limits_{x\to 0}\frac {2\sin(x)-\arctan(x)-x\cos(x^2)}{x^5}$, using power series?

How to evaluate the $\displaystyle\lim\limits_{x\to 0}\frac {2\sin(x)-\arctan(x)-x\cos(x^2)}{x^5}$, using power series? It made sense to first try and build the numerator using power series that are ...
3
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2answers
30 views

Inverse Rule for Formal Power Series

I am just really starting to get into formal power series and understanding them. I'm particularly interested in looking at the coefficients generated by the inverse of a formal power series: ...
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2answers
55 views

Example of a Power Series Given Interval of Convergence

This was a thought question assigned to our calc II class, and I wasn't sure how to approach it. Give an example of a power series whose interval of convergence is $(0, \frac{4}{3}]$. Show ...
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2answers
34 views

Power series properties. The sum of two series.

Let $\sum a_nz^n$ and $\sum b_nz^n$ power series with radiuses of convergences $R_1,R_2$ respectively. Suppose the radius of convergence of $(\sum a_n+b_n)z^n$ is $R$. Find an example in which ...
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1answer
97 views

Power series related to Bernoulli numbers

I'm reading Tenenbaum's Introduction to analytic number theory. He defines Bernoulli polynomials as the unique sequence $B_n$ such that $B_0=1$ $\forall n\geq0, B_{n+1}'(X)=(n+1)B_n(X)$ ...
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1answer
20 views

Having Trouble finding a simplified power series representation.

Partial fractions seemed the most efficient route to take. However, I am having trouble at the end.
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1answer
34 views

Power series with coefficients in primary rings

Let $P$ be a prime ideal in a commutative ring $R$ with unity such that an ideal $Q$ is $P$-primary and some power of $P$ is a subset of $Q$. I want to show that $\sqrt {Q[[x]]}=P[[x]]$. If a ...
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1answer
44 views

What's the sum of the series? $\sum_{n=1}^\infty \frac{ (-1)^{n+1}x^{2n+1}}{(2n-1)(2n+1)}$

So, for $$\sum_{n=1}^\infty \frac{ (-1)^{n+1}x^{2n+1}}{(2n-1)(2n+1)}\ ,$$ I know the answer involves $\tan^{-1}(x)$ because the series representation is there for it. But, I don't know what to do ...
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2answers
49 views

What is the power series for the function $\ln(1+x^3)$? [closed]

How do you find the power series for the function $\ln(1+x^3)$?
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0answers
77 views

40th derivative of a function

I would like to have some verification to see if my answer is correct. The given function is $f(x)=ln(1+x^2)$ and I need the 40th derivative at $x=0$. Here is my work: Using series one can manipulate ...
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1answer
94 views

Some issues with proving that a sequence is convergent

I recently tried (in the sense that I believe the thesis holds) to prove that, given $a\in\mathbb{R}^+$, there exists $$\lim_{n\rightarrow+\infty}\sqrt[n]{\sum_{k=0}^{\lfloor n/5\rfloor}{n-4k\choose ...
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1answer
54 views

Poisson power series

We have a Poisson power series of $$Y=\sum\limits_{k=0}^{\infty}e^{-\pi\lambda v^2}\frac{(\pi\lambda v^2)^k}{k!}(A)^k $$ If we have a disk with radius $v$ where A is defined as the density of a ...
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1answer
24 views

Representing functions as power series and finding $c_0,c_1,c_2$…

I have a problem with representing functions as power series: I was trying to find $c_0$, $c_1$, $c_2$, ... and the Radius of Convergence but I'm not sure how to do this, I'm a bit lost here could ...
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0answers
15 views

On algebraicity of a formal power series

In his paper "Noncommutative identities" M. Kontsevich states the following: Theorem 2. For any $P=P(x,y)=1+\cdots\in\mathbb{C}[x,y]$ expand ...
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1answer
25 views

Power (Maclaurin) series of $a/x ( exp(x/a) - 1)^{-1}$

I'm currently stuck on an explanation that works with a power series expansion of: $ \frac{a}{\eta}\: \frac{1}{e^{\frac{\eta}{a}}-1}$ As you can see, the function diverges around $\eta=0$, which is ...
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2answers
57 views

Expected number of die tosses to get something less than 5

Assume you have a die in your hand. Each time you throw it, you look to see what value you would get. If the value is greater than 4, roll the die again. Otherwise you stop. Let X be the number of ...
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1answer
30 views

Estimating a power series for the order of an entire function

Let $0<s<1$ and consider the power series $$\sum_{n=0}^{\infty}\frac{r^n}{(n!)^{1/s}}.$$ I need to show that for any given $\epsilon>0$, there exists $R>0$ such that for all $r>R$, ...
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0answers
20 views

Why would you choose the Method of Frobenius over a Power Series solution to solve a DE?

I'm trying to determine where it would be more appropriate to use one or the other. To further clarify: Where would it make more sense to use: $y=\sum_{n=0}^{\infty}c_n(x-x_0)^{n+r}$ instead of ...
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3answers
60 views

Sum of power series using derivation or integration

could anyone help with this question? $$\sum_{n=1}^{\infty}\frac{(x-\frac{1}{2})^{n+1}}{n(n+1)}$$ I have to find sum of this power series using differentiation or integration. Thanks a lot!
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0answers
34 views

Does a better form exists for the coefficients of this product of power series?

Let $$f(a,b,t) = \sqrt{1-at}\sqrt{1-bt}$$ We take the series for $\sqrt{1-at}$ and $\sqrt{1-bt}$ around $t = 0$ and multiply them together to find $$f(a,b,t) = \sum_{n=0}^\infty ...
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2answers
36 views

Sum of powers of 2 from 1 to log(N).

I came across the following sum: $\sum_{m=1}^{\log_2(N)} 2^{m}$. My intuition tells me that this should be bounded by 2N, but how would I prove this?
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2answers
71 views

Prove these two functions have the same coefficients

We have $\displaystyle p(x) = \frac{1}{1-x}\cdot \frac{1}{1-x^3} \cdot \frac{1}{1-x^5} \cdot \ ...$ and $\displaystyle q(x) = (1+x)\cdot(1+x^2) \cdot (1+x^3)\cdot \ ...$. Let's say that these two ...
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1answer
27 views

How to formulate the product of two generating functions without their final terms?

I know that if we have two generating functions like so: $A(z) = \sum_{n=0}^\infty a_nz^n$ and $B(z) = \sum_{n=0}^\infty b_nz^n$ Then we can write $A(z)B(z) = \sum_{n=0}^\infty(a_0b_n + a_1b_{n-1} ...
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3answers
68 views

Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method

Use series methods to solve: $(x^2 + 1)y'' - 6xy' + 10y =0$ a) Give the recursion formula b) Give the first two non-zero terms of the solution corresponding to $a_0 = 1$ and $a_1 = 0$ ...
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1answer
56 views

Use series methods to find solution corresponding to..

Use series methods to find solution corresponding to $a_0 = 1$ for the equation $(x+1)y' - y = 0$ Here is my work. Can someone verify that I have the correct solution: So for my final solution I ...
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1answer
26 views

Changing order of summation including a min in the summation

Lets say I have the following expression: $$ h(x) = \sum_{k=1}^n \sum_{v=1}^{\min\{k,j\}} \frac{(-1)^{n-k}k!}{(k-v)!} {n \brack k}f(x)^{k-v} B_{n,v}^f(x) $$ Now my goal is to have the $v$ ...
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0answers
42 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of ...
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1answer
71 views

$\lim\sum_{k=0}^{\lfloor\delta n\rfloor} \frac{n^k}{k!}e^{-n}$ and Poisson distribution

Problem: Let $X_1,X_2\ldots$ be some independent random variables with Poisson distribution with parameter 1. Show that for every $\epsilon > 0$ sequence $S_n-(1-\epsilon)n$ diverges to $\infty$ ...
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1answer
44 views

How would I set up the Taylor's Inequality to prove that the function is equal to its Taylor Series expansion?

How would I set up the Taylor's Inequality to prove that the function $f(x) = \frac{1}{x}$ is equal to its Taylor Series expansion centered at $x=1$? I've done the Taylor series expansion, but ...
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1answer
31 views

One of the points of interval of convergence of the power series $\sum_{n=0}^\infty \left(\frac{x^8-1}{3}\right)^n$ is not a real number?

One of my points is not a real number in my interval of convergence! This is how I calculated it: $$\lim_{n\to\infty} \left|\frac{x^8-1}{3}\right|$$ Using root test $\frac{x^8-1}{3} <1$ ...
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6answers
231 views

How to Find the Function of a Given Power Series?

(Please see edit below; I originally asked how to find a power series expansion of a given function, but I now wanted to know how to do the reverse case.) Can someone please explain how to find the ...
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4answers
127 views

Write a function as $\sum _{n=0} ^{\infty} a_n x^n$

We have $f(x) = (x+ x^2 + x^3 + x^4 + x^5 + x^6)^4$. Now I want to write this as $\sum _{n=0} ^{\infty} a_n x^n$. What I got: $f(x) = (x+ x^2 + x^3 + x^4 + x^5 + x^6)^4 = x^4 (1+ x + x^2 + x^3 + ...
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2answers
34 views

Find a recurrence relation and solve it

Let $a_n$ be the nummber of ways that 4 people can throw $n$ eyes together with a die. Every person throw once. Now I want to find a generating function and compute $a_n$ for different $n$. To do ...
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1answer
22 views

Finding interval of convergence for complicated sum

I'm going through old exams for my Calc III course and came across a problem that I did not know how to do. The problem is: Find the interval of convergence of the series ...
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1answer
30 views

Stuck on finding where $\sum_1^{\infty} (x+4)^{n}$ converges conditionally.

I have all the questions correct on my hw except for one: find where $\sum_1^{\infty} (x+4)^{n}$ converges conditionally. Radius of Convergence I got 1 for this, by using the root test and finding ...
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1answer
35 views

Find series representation of a function

Let $\displaystyle f(x) = x^4 (\frac{1-x^6}{1-x})^4$. Now I want to find the series representation of this function. Note that $\displaystyle \frac{d^3}{dx^3} (\frac{1}{1-x})= \frac{6}{(1-x)^4}$. ...
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3answers
75 views

Compute the sum $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$

Do you know how to find the limit of $\sum_{k=1}^{\infty}k^mz^k$ where $|z|<1$ and m is a natural number? I've tried to google it in wiki but I do not understand the closed form ...
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1answer
51 views

Is a holomorphic function analytic in a ‘real’ sense?

I am taking a course in complex analysis, and I asked myself the following question: If a function $ f: \mathbb{C} \to \mathbb{C} $ is holomorphic, can its real and imaginary parts be given by a ...
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0answers
30 views

Power series writing terms

Can someone explain how I should be solving for theses terms? I got my series to be $10(-7x)^n$, but I dont know what to do from there. Original link to image: http://imgur.com/6sUVvaR
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2answers
46 views

Transformation of a function into a power series [closed]

How can I transform the real functions $\frac{1}{1-\sin x}$ and $\frac{x}{e^x-1}$ into power series with $x_0=0$?
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1answer
40 views

Poles of power series

This may be a trivial question, but I haven't been able to find an answer. Given a power series about $x_0$ $F(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$, how do we find its (complex) poles? What about the ...
2
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0answers
25 views

Formal Expansion of another Expansion

Given a function $f(x)=\sum_{n=1}^{\infty}\frac{c_n}{x^n n!}$, where $c_n$ are constants, we want to find the formal series expansion of the function $g(x)=\exp(f(x))$ in terms of $x$. I want to ...
1
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1answer
32 views

Power Series: $\sum_1^{\infty} (x)^{n}\frac{n^3}{n!}$

I just started learning about the power series, can someone help me with finding the radius of convergence and interval of convergence? So I am stuck on the radius of convergence because apparently I ...