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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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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36 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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69 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
310 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
42 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
112 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
232 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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75 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
24 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
39 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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29 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 ...
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0answers
80 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
55 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
89 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
31 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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1answer
129 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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51 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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63 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
33 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 $s_4(x)$ ...
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513 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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136 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
42 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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271 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
40 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
31 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
75 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
31 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 ...
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172 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
67 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
26 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
50 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
70 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
52 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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1answer
102 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. ...
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1answer
51 views

Large-z limit of the *other* second derivative of the Laguerre polynomial

I'm trying to find the asymptotic behavior of the second derivative of the Laguerre polynomial (more precisely, the associated analytic function), $\frac{\partial}{\partial n^2}L_{n}(z)$, as $z\to ...
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2answers
62 views

Estimating integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$

Estimate the definite integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$ with an error of at most $10^{-4}$, using the Alternating Series Estimation Theorem. My approach is as follows: I found the ...
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2answers
55 views

Power Series: Derivative

Given a Banach space $E$. Consider a series: $$|t|\leq R:\quad\sum_{k=0}^\infty A_k t^k\quad(A_k\in E)$$ Is there an elegant proof of: $$\left(\sum_{k=0}^\infty A_k t^k\right)'=\sum_{k=0}^\infty A_k ...
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0answers
49 views

Find power series expansion and radius of convergence

I have to find power series of $\frac{1}{(z-1)(z-2)}$ centered at $3+i$ and give its radius of convergence. I just simply transformed it using partial fractions and geometric series expansion and ...
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106 views

How do I express the sum $(1+k)+(1+k)^2+(1+k)^3…+(1+k)^N$ for $|k|<<1$ as a series?

Wolfram Alpha provides the following exact solution $$ \sum_1^N (1+k)^i = \frac{(1+k)\,((1+k)^N-1)}{k}.$$ I wish to solve for $N$ of the order of several thousand and $|k|$ very small (c. $10^{-12}).$ ...
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1answer
48 views

Find Laurent series for $\frac{(z+1)}{z(z-4)^3}$ in $0<|z-4|<4$

Find Laurent series for $\frac{(z+1)}{z(z-4)^3}$ in $0<|z-4|<4$. First we perform partial fraction and we get: $\frac{A}{z}+\frac{B}{z-4}+\frac{C}{(z-4)^2}+\frac{D}{(z-4)^3}$. My first ...
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1answer
29 views

Find infinite set for which the series diverges

I'm looking to clarify the meaning of a question, and would greatly appreciate any feedback. Given a function $f_n(x)$, I am to construct an infinite set S such the series ...
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3answers
130 views

Power series expansion involving non integer exponent

I'm working on a real and complex analysis course right now and one power series question has me really stumped: I'm not sure what to do with the non integer in the exponent, as my initial plan of ...
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1answer
33 views

Writing ODE solution as sun of power series?

Let $f(t), g(t)$ be polynomials, and let $y$ be a function of $t$. Given the ODE $y'' + f(t) y' + g(t) y = 0$ with initial conditions $y(0) = \alpha$ and $y'(0) = \beta$, write $y$ as the sum of a ...
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102 views

For which values of x does the power series converge or diverge?

First, I know that the series converges when |x+2| < R and diverges when |x+2| > R. So now I have that the radius of convergence is somewhere between 2 and 3. However, this doesn't really give me ...
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3answers
150 views

Questions on the differential equation $df/dx=-[f(x)]^2$

I have another group project problem I am having trouble with. Here is the first part of the problem: "Consider the differential equation $df/dx=-[f(x)]^2$, with initial condition $f(0)=a$." ...
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1answer
25 views

General Case of Convergence of a Power Series

The Question If $f(x) = \sum c_nx^n$, where $c_{n+4} = c_n$ for all $n\ge 0$, find the interval of convergence of the series and a formula for $f(x)$ My Work and Question I haven't been able to do ...
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487 views

Calculating the Laurent Series of $\tan z$

I need help calculating the laurent series of $\tan z$ around the points $z=0$, $z=\pi/2$, and $z=\pi$. How would one go about doing this? I solved an almost identical question that was "Derive the ...
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1answer
25 views

If $S_n\to \infty$ as $n\to \infty$ is the following inequality valid or when is it valid? $\frac{a_n}{S_{n-1}}\leq \frac{C}{n}.$

Let $a_n$ be a sequences of positive real numbers and $S_n=\sum_{k=1}^{n}a_k.$ If $S_n\to \infty$ as $n\to \infty$ is the following inequality valid or when is it valid?$(C>0)$ ...
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3answers
63 views

Find the interval of convergence of $x + \frac{1}{2} x^2 + 3x^3 + \frac{1}{4}x^4 +…$

How to find the interval of convergence of the following series: $x + \frac{1}{2} x^2 + 3x^3 + \frac{1}{4}x^4 +...$ I have no idea what to proceed. Any help? Thanks!
2
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2answers
144 views

Calculate Laurent Series for $\frac{\ln z}{(z-1)^3}$ about $z=1$

Calculate the Laurent series of the function $g(z)= \frac{\ln z}{(z-1)^3}$ about the point $z=1$. Well since the singularity and the centre of the circle we are expanding about collide, I can just ...