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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Question on the sum $\sum_{n=1}^{\infty}\frac{x^n}{n} = -\ln(1-x)$

$f(x) = \displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n} = x + \frac{x^2}{2} + \frac{x^3}{3} + ... = -\ln(1-x)$ for $|x| < 1$. $f'(x) = \displaystyle\sum_{n=1}^{\infty}x^{n-1} = 1 + x + x^2 + x^3 ...
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1answer
25 views

Uniform convergence of series and continuity of $f$

This is from Ross's Elementary Analysis Textbook: The series $(2^{-n})(x^n)$ from $n=1$ to $n= \infty$ represents a continuous function on $(-2,2)$, but the convergence isn't uniform. He points out ...
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2answers
47 views

Integral with series

How do I represent this integral $$\int_{0}^{1} \frac{10}{10+x^4} dx$$ as a series so that I can calculate with an error of less than $10^{-5}$.
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0answers
13 views

Gradient inequality in a simpler case

Let $f : \mathbb{R} \to \mathbb{R}$ be a analytic function. There exists $\theta \in (0,1/2]$, $c$, $\sigma$ such that for every $|x-a|\le \sigma$ $$ |f(x) - f(a)|^{1-\theta} \le c |f'(x)|. $$ This ...
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0answers
28 views

how to expand $\exp(-b x)$ ($b>0$) when $1<x<B$?

Let $b>0$, how to expand $\exp(-bx)$ when $1<x<B$? I am seeking a series expansion like the following ($n_0> 0$): $$\exp(-bx)=\sum_{n=n_0}^{\infty}\frac{a_n}{x^n} $$ EDIT: ...
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3answers
83 views

How to find the coefficient of a power in a power series??

How can I find the coefficient of $x^{80}$ in the power series $$(1+x+x^{2}+x^{3}+x^{4}+\cdots)(x^{2}+x^{4}+x^{6}+x^{8}+\cdots)(1+x^{3}+x^{5})\,?$$ Is there a general method to this?
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1answer
283 views

Why are there two series representations of the natural logarithm?

On the Wikipedia article of the natural logarithm one finds two different series representations for $\ln(x)$: $\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$ ...
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1answer
11 views

fast way to find order of a (semi) geometric series

Is there a fast way to find whether the order of the following is $O(T^2)$ or $O(T)$? I've been trying to find the exact thing by using the geometric series multiple time, but it is so lengthy and ...
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1answer
147 views

I have a question about integrating, and what to do about the constant. $\displaystyle\int\frac{1}{1-z}dz$

http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf On page 44, they conclude that $g'(z) = - \displaystyle\frac{\ln(1-z)}{z}$ by saying that it is just ...
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1answer
50 views

Power series $e^{-x^2}$

How would I create a power series of $f(x)=e^{-x^2}$ around $x_0=1$ without using a Taylor series? I need to know this for my upcoming exam so I would be really grateful to anyone who could show me ...
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2answers
31 views

Calculate $a_n$ with formal power series

I have $$A(x) = \sum_{n=0}^\infty a_n x^n$$ and $$A(x) = (1+x)/(1+7x+6x^2)$$ I need to find $a_0,a_1,a_2,a_3$. I multiply on each side and I get $$ (1+7x+6x^2) (a_0+a_1x+a_2x^2+a_3x^3+\ldots)=1+x ...
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1answer
21 views

Variables, Square roots, and exponents

Answer : $x^2$ I got $x^n$, shouldn't I be multiplying the variables in the parentheses first. Thus cancelling out the roots and left with $x$ then to the power of n? thus -> $x^n$ ? Please explain ...
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0answers
21 views

How to simplify this series $\sum_{m=0}^{\infty}s^m{n+m-1 \choose m}p^n (1-p)^m$ for m $\ge$ 0

I came across this is my textbook and I was wondering if anybody has a trick to simplify this series $\sum_{m=0}^{\infty}s^m{n+m-1 \choose m}p^n (1-p)^m$ for m $\ge$ 0 $$= ({\frac{p}{1-s(1-p)}})^n$$ ...
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1answer
29 views

Basic Geometric Series Question

Calculation of $ \sum_{n=0}^{\infty}2^{2n} z^{2n} $ The answer is We note that the n-th summand has the form $(2z)^n$ Denoting w = 2z The sum is sigma of 0 to n summand being $(w)^n$ which can be ...
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2answers
27 views

ordinary generating function of some sequence

What is the ordinary generating function of the sequence whose general term is $a_n = {n+k \choose k}$?. I cannot find it in the list given in the book generatingfunctionology, by Herbert S. Wilf. Is ...
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1answer
39 views

Finding the Accuracy of a Taylor Polynomial for the Approximation $f(x) \approx T_{n}(x)$

Let $$ f(x) = \sin(x), \quad a = \frac{\pi}{6}, \quad n = 4, \quad 0 \leq x \leq \frac{\pi}{3} $$ Find a fourth degree ($n=4$) Taylor polynomial for $f$. $$ T_{4}(x) = ...
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1answer
30 views

Exponential series is cosh(x), how to show using summation?

I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} ‎\frac{(x)^{2n}‎}{(2n)!}‎ $$ I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that ...
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1answer
29 views

Power series with $x^{4n}$

I'm new to this Forum. I do not find an approach to solve the following problem (from the book "Herbert Wallner, Aufgabensammlung Mathematik Band 1", so this is not a homework question): For which ...
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3answers
105 views

Theres a small detail in this proof on why $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2} = \frac{\pi^2}{6}$ that I cant figure out

http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf Here is a link to the article I have been reading. Its really interesting and easy to follow. What bothers me is a result ...
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1answer
42 views

How to know if two power series solutions are linearly independent?

I'm currently studying power series to solve differential equations. I would like to know if there's a way to tell whether two solutions are linearly independent or not. I think evaluating the ...
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2answers
26 views

coefficients for terms of power series

I'm asked to represent the function $\displaystyle \frac{2 x}{10 + x}$ as a power series $f(x) = \displaystyle \sum_{n=0}^\infty c_n x^n$ I found this to be $\displaystyle \sum_{n=0}^\infty ...
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1answer
19 views

Differential equation by power series with IVP not at $0$.

I was wondering whether one could solve a differential equation with initial value at $x_0\neq 0$. I think that the series must converge for $x=x_0$, otherwise the IVP wouldn't make sense. It seems as ...
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2answers
35 views

Convergence of an infinite series?

I feel the coefficient Cn has to be zero in order for the original series to converge, as the power series of 4^n will diverge as n - > ∞. Are there any other ways for this series to converge, and ...
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1answer
27 views

Functions in $\mathbb {R}[X] $

For the ring of polynomials over the reals, which can be considered an infinite-dimensional vector space with infinite monomial basis, is the following true: Any analytic function $f$, which is ...
2
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0answers
43 views

Second order differential equation, power series method

Solve the differential equation $$(x+2)y''-xy'+(1-x^2)y=0 ; \quad X_0=1$$ using the power series method about the point $x_0=1$. I get to this step after deriving the derivatives of the ...
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0answers
39 views

Is $P(x)+\sum a_nx^n$ still valid power series notation?

Would adding any arbitrary polynomial to a power series satisfy the conditions for a power series? Example: $\frac{1}{1-x}$ has the power series, $\sum x^n$. Would $1+\sum x^n$ still be a power ...
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1answer
26 views

Complex power series - Radius of convergence

Let $f$ be analytic in the unit disk. Then we can write that as, $$f(z)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}z^n,|z|<1$$ Now let $a_n=\frac{f^{(n)}(0)}{n!}$. So the radius of convergence $R$ is ...
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1answer
25 views

How to determine if there exist at least one number that is generated by both of the given generating functions?

I'm just learning about Generating Functions so my question might not completely make sense (in that case, I apologize). I want to know whether there exist at least one number that is generated by ...
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2answers
35 views

Convergence of complex mercator series

I'm trying to find out for which $|z|=1$ the series $$\sum_{n=1}^\infty{}\frac{z^n}{n}$$ converges. It diverges for $z=1$ (harmonic series) and converges for $z=-1$ (alternating harmonic series). I ...
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37 views

$\sum\limits_{k=1}^{n^2}E(\sqrt{k})\quad n\in\mathbb{N}^{*}$

I found in my archives solution of this exercise Calculate $$\sum\limits_{k=1}^{n^2}E(\sqrt{k})\quad n\in\mathbb{N}^{*}$$ E represent the floor function Solution: they made Let ...
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2answers
46 views

If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)

Prove that if $f$ is continuous on $\mathbb R$ and $$\lim_{x \to +\infty} [f(x+1)-f(x)] = l,$$ then $$\lim_{x\to +\infty} f(x)/x =l.$$ So I've been trying for hours to use the series ...
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1answer
26 views

What is $s_3$ and $s_4$ for $x$

$\sum_{i=0}^n i^k = s_k(n)$, $s_k$ polynomial from degree $k+1$ I have already shown for $s_2(x) = \frac{x(x+1)(2x+1)}6$ How from the sum and $s_2(x)$ can be shown for $s_3(x)$ and $s_4(x)$ ...
4
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2answers
315 views

Graphical interpretation of infinite power series?

Can someone please give me a graphical interpretation/sense of infinite power series? Some functions such as exponentials, sines, and cosines are infinite power series, but what does that mean and ...
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2answers
92 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
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1answer
28 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I+x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
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2answers
167 views

Finding Sum for Infinite Series

Normally when I keep try multiple ways to solve a problem, I get an idea of where to start, and eventually can solve it. But it hasn't been the way for this question and I've been stuck for hours. ...
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1answer
35 views

why are these two power series the same

$$-\sum_{\color{red}{n=1}}^{\infty}nc_{n}x^{n}=-\sum_{\color{red}{n=0}}^{\infty}nc_{n}x^{n}$$ How come one starts at $1$ and the other starts at $0$ yet their equal? Do they both equal infinity?
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1answer
28 views

Where the power series is convergent

Where $f(z)=\sum_{n=1}^{\infty}\frac{(2i)^n}{n}z^n$ is convergent? I checked that the radius of convergence is equal to $\frac{1}{2}$. Now, since we know that the series ...
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1answer
35 views

Laurent Series Expansion for $f(z)=\dfrac{z+2}{(z+1)(z-2)}$ in $\{1<|z|<2\}$ and $\{2<|z|<\infty\}$

I'm trying to get the Laurent Series expansion of the function stated in the title in the stated regions. My approach is as follows: We can first break up $f(z)$ using partial fractions ...
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1answer
24 views

If $x_k ≥ 0\;\forall \in \mathbb N$, and $y_k$ a bounded sequence, then the series $\sum_{k=1}^\infty x_ky_k$ converges

Hi I'm really struggling with this proof. For a start I'm struggling to believe it's true: For example, if we take $x_k = \dfrac{1}{k^2}$ and $y_k = -k^3$ (which is bounded above by any positive ...
4
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2answers
147 views

the first $2k$ terms of the power series of $\sec x + \tan x$ at $x=-\pi/2$

We know the power series of $\sec x+\tan x$ is as follows, $f(x)=\sum_{n\geq 0}\frac{E_n}{n!}x^n$, where $E_n$ is Euler Zigzag numbers and clearly the radius of convergence of $f(x)$ is $\pi/2$. ...
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1answer
40 views

Sum of Series - Intelligent Manipulation

I have been learning about sums of series, and am very curious: If we know that $e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$ What is the value of the following power series: ...
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2answers
22 views

Formal power series question

$$(1-t)^d \sum_{k = 0}^{\infty} \binom{d+k-1}{d-1} t^k = 1$$ How can this be proven? Thanks in advance.
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1answer
43 views

Maclaurin series construction

I am asked to find the Taylor (Maclaurin) series for $9xe^x$ at $x=0$. I did the following: $f(x)=9xe^x \implies f'(x)=9e^x(1+x) \implies f''(x)=9e^x(2+x)$ et cetera. This yields: $P_0(x)=0, ...
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0answers
12 views

How to determine singularities of a series?

Given a double Fourier series, how do we determine its singularities ? PS: I wonder how we find singularities(mathematically) if a function cannot be expressed in a closed form.
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0answers
55 views

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent. I use a root test: $\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=\lim_{n\rightarrow\infty} ...
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1answer
48 views

Coefficeient of $x^k$ in $(1+x)^n$ when $n<0$

I know this is a very basic question. But I simply cannot derive the final answer. We have the alternate form of binomial theorem if we want to deal with negative exponents. ...
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0answers
19 views

To find sum of series involving combinations and show convergence of series

to make sum of series including combinations ${N\choose 1}{N\choose 0}+{N \choose 2}{N\choose 1}a^2 b^{-2} + {N\choose 3}{N \choose 2}a^4 b^{-4}+{N\choose 4}{N \choose 3}a^6 b^{-6}+...$Is it possible ...
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1answer
53 views

Solving the ODE $y''-2x^2y'+4xy=x^2+2x+2$ using power series

I am trying to solve this nonhomogeneous ODE: $$y''-2x^2y'+4xy=x^2+2x+2$$ I know it's a power series, but when I get down to the very end, I end up with a $C_0$ term, a $C_1$ term, and a $C_2$ term. ...
0
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0answers
36 views

What are the possible expansions for $f(z)$ at $0$ for disks and annuli?

For the expression $f(z)$ what are its all possible expansions (I am considering disks and annuli) around the origin and where do they converge? $$ f(z) = z + 2z^2 + 3z^3 + \ldots + nz^n + \ldots = ...