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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21 views

Second Order Linear Non-Homogeneous DE solution with Power Series $x^2y'' - 4xy' + 6y = x^2 \cos x$

My instructor wants me to solve the above equation using power series and another method, and then to confirm the results are the same This equation does not have constant coefficients and a can't ...
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1answer
42 views

Help with generating functions

I've got two questions. I'm trying to extract the "coefficients" of a power series. I think my terminology is incorrect here but here is what I mean. Here are some examples A(Z) = 1/(1-Z) ...
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2answers
43 views

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series. I know that $e^{x^2}=\sum_{n \geq0} \frac{x^{2n}}{n!}$, but I don't know how to find the coefficients $a_n$ in considering ...
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1answer
35 views

Frobenius Method to Solve a Differential Equation

Having the equation $$x^{2}y''+xy'+x^{2}y=0$$ I get the indicial equation at get r=0, and am left with the equation. ...
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2answers
57 views

$(x-x_0)^0$ in power series [duplicate]

When I first studied power series in high school, the teacher gave the following general definition: \begin{equation} f(x)=\sum_{n=0}^{\infty}a_n (x-x_0)^n \end{equation} He then proceeded to ...
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2answers
33 views

Difficulty finding a power series representation

I have to find a power series representation and interval of convergence for $$f(x) = \frac{x-x^2}{(1+2x)^3}$$ Noting that $\frac{1}{1+2x}=\frac{1}{1-(-2x)}=\sum_{n=0}^\infty(-2x)^n$, I start taking ...
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1answer
20 views

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$.

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$. Now I know that $b_n=\dfrac{f^{(n)}(5)}{n!}$. I have tried various things but I think there is something wrong with my ...
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1answer
36 views

Proving that a function is real-analytic

I try to solve the following exercise: Let $f:\mathbb{R}\to\mathbb{R}$ with $f(x):=\frac{1}{1+x^4}$. Prove that $f(x)$ is real analytic and compute the radius of convergence of it's Taylor series at ...
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2answers
43 views

Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that ...
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1answer
34 views

How to expand the summation term with power?

How to expand the following: $$ \left( \sum^{M}_{m=0} \frac{x^{m}}{m!} \right)^{K} $$ where $M$ and $K$ are positive integers.
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2answers
193 views

Is it true that $log(i) = \frac\pi2i$ ? If so, are both of these legitimate proofs? They seem too beautiful not to be…

Sorry if this is a naive question. I have not yet taken any upper level math courses involving complex numbers. However, in preparation for those courses, together with utilizing the knowledge that ...
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1answer
35 views

Power series solution to $y' = y(1-y)$

Find the first five terms of the power series solution to the differential equation: $$y' = y(1-y)$$ Letting $y = a_0+a_1x+a_2x^2+a_3x^3+...$ It's evident that: $$y' = \frac{dy}{dx} = ...
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1answer
50 views

Show that $e^{\varepsilon |x|^{\varepsilon}}$ grows faster than $\sum_{k=0}^{\infty} {|x|^{2k}}/{(k!)^2}$

I am wondering whether we have for $$f(x):=\sum_{k=0}^{\infty} \frac{|x|^{2k}}{(k!)^2} $$ that $$\lim_{x \rightarrow \infty} \frac{e^{\varepsilon |x|^{\varepsilon}}}{f(x)} = \infty$$ for any ...
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1answer
54 views

Taylor expansion of $\frac1{\|x-y\|}$

Let $0\neq y\in \mathbb{R^3}$ define a function $f$ on $\mathbb{R^3}$ as $$ f(x) = \frac1{\| x-y\|} $$ What are derivatives of $f$ in zero? Or equivalently, what is the Taylor series of $f$ at ...
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1answer
30 views

Expansion of $f = \sum_{n=1}^\infty \frac{1}{2^n} \frac{z^n}{1 - z^n}$ in power series around $z = 0$

Let $f = \sum_{n=1}^\infty \frac{1}{2^n} \frac{z^n}{1 - z^n}$, for $z \in \mathbb C \setminus ${$z \in \mathbb C: \exists n \geq 1,\quad z^n = 1$}. By the ratio test, the series converges when ...
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1answer
35 views

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$.

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$. Find a closed form expression for all x which converge and hence evaluate ...
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2answers
18 views

Why is it that the interval of convergence is half open?

I am given the following power series and asked to find the radius of convergence and determine the exact interval of convergence $$\sum\biggr(\frac{3^n}{n\cdot 4^{n}}\bigg)x^n \Leftrightarrow ...
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2answers
24 views

What is the power series and domain for this function?

$$f(x)= \frac{x}{1+5x^2}$$ I got the power series: $$\sum_{n=0}^\infty (-1)^n (5^n)(x^{2n+1})$$ Assuming this is correct I would think the domain would be $$(-5^{1/3}, 5^{1/3})$$ because the absolute ...
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2answers
51 views

Show that a power series is analytic inside its radius of convergence

Let $f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k$ with radius of convergence $R$ then $f$ is analytic on the open disk around $z_0$ with radius $R$. What I was thinking about is an approach based on this ...
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0answers
20 views

Continuity of series implies continuity of coefficients?

For each $t\in [0,1]$ let $f_t(z)$ be an entire function. By holomorphicity it equals its own Taylor series: $$f_t(z) = \sum_{n=0}^\infty a_n(t)\,z^n, \qquad \forall \,z\in\mathbb{C}.$$ Now ...
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29 views

Solving a power series centered at 0 by integrating another power series

Problem: Find the power series for g(x) centered at 0 by integrating the power series f(x). Give answer in sigma notation along with the first four nonzero terms. $$g(x)=\ln{(1-3x)}\space \land ...
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1answer
31 views

Power series expansion of a complex function

How can I depict function $f(z)=\sqrt{1+\sqrt{1+z^2}}$ as a power series around zero? Where Log is a function going from $C \setminus (-\infty,0]$? Since with such a logarythm $Re(\sqrt{z}>0$ then ...
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2answers
58 views

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$?

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$? So I think we expand to $x^2$ since the lowest term for $\ln(1+x)$ is $x$ Let $u=\arctan{(x)}$ ...
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1answer
10 views

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$?

How to compute taylor series of $\sin{x}$ about $a=\frac{\pi}{4}$? We know $\sin{x}=\sum^{\infty}_{n=0}\frac{(-1)^nx^{2n+1}}{(2n+1)!}$. Let $t=x-\frac{\pi}{4}$, then $t+\frac{\pi}{4}=x$ Then ...
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1answer
37 views

Laurent Series of $(z-2)/(z+1)$ at $z=-1$ [closed]

What's the Laurent series expansion of $\frac{z-2}{z+1}$ at $z=-1$?
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2answers
80 views

Verify f'(x) = e^x

The following is a proof I wrote to prove that given $f(x)=e^x$, $f'(x)=e^x$. For this proof we must use the Taylor Series for $e^x$, $\sum\limits_{n=0}^{\infty}\dfrac{x^n}{n!}$. Since the derivative ...
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1answer
17 views

Criteria for convergence of power series

Given the power series: $\; \sum_{i=0}^{\infty}a_nz^n \;$ Proof that if there exist $s,M \in \mathbb R $ such that $|a_n|s^n \le M$ then the power series converges for every $|z|\lt s$ Can someone ...
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1answer
22 views

How do I find a power series for this function?

Given the function: $$f(x) =\frac{(11+x)}{(1-x)}$$ how would I find a power series representation? I started by rewriting the function as $$(11+x)\frac{(1)}{(1-x)}$$ and then arrived at ...
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1answer
35 views

power series find values for $\sum_{n=1}^{\infty}\frac n{2^n}$ and $\sum_{n=1}^{\infty}\frac {n^2}{2^n}$

Hi I am in a basic real class and I am confused about the question: Given the geometric series: $$\frac 1{1-x}=1+x+x^2+x^3...$$ for all $|x|<1$ use results about the power series in this section ...
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2answers
54 views

Finding Exact Values of Specific Infinite Series

Prove that $\Sigma_{n=1}^{\infty}(n/2^n)=2$ and that $\Sigma_{n=1}^{\infty}(n^2/2^n)=6$. Thoughts: I have a feeling that if someone shows me how to do one, I'll be able to figure out the other. So ...
2
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1answer
25 views

How to compute the following series using taylor expansion manipulation?

How to compute $\sum^{\infty}_{n=0} \frac{x^n}{(n+2)n!}$ and $\sum^{\infty}_{n=0}(-1)^n \frac{(n+1)x^{2n+1}}{(2n+1)!}$ using taylor expansion manipulation? $1.\sum^{\infty}_{n=0} ...
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3answers
52 views

Complex power series expansion of $\frac{e^z}{1+z}$

I'm trying to find complex power series expansion of $\frac{e^z}{1+z}$ centered at $z=0$ and its radius of convergence. Here is my attempt: Since $e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$, we can ...
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0answers
15 views

Maclaurin polynomial expansion of $y$ about 1?

Consider the differential equation $\frac{dy}{dx}=2x+\frac{y}{x}$, where $\frac{dy}{dx}=1$ when $x=1$. Find the first three non-zero terms in the Maclaurin polynomial expansion for $y$ about ...
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2answers
44 views

Finding the power series of a complex function

So I have the function $$\frac{z^2}{(z+i)(z-i)^2}.$$ I want to determine the power series around $z=0$ of this function. I know that the power series is $\sum_{n=0}^\infty a_n(z-a)^n$, where ...
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1answer
18 views

Is a convergent power series on an open set continuous on that set?

Question in the title. If a power series $f(x)$ is pointwise (or if this is too weak, uniformly) convergent for every $x$ in an open set $U$ in the reals, is it a continuous function of $x$?
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1answer
48 views

Finding the sum of this series $\sum (\alpha x)^n$

I'm looking for help on how to find the sum and interval of convergence of this series (Starts at 0 and goes to infinity). Now this one is giving me trouble because I've never seen a series with the ...
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7answers
549 views

How do you create an alternating series with the sign being the same twice in a row?

I am working on a Taylor series question and I have created a series which alternates however, it does so in doubles. in other words it follows the following pattern: $x$, $x$, $-x$, $-x$, $x$, ...
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1answer
57 views

Show L'Hospital limit for exponential function and power series

Given a series $$f(t):=\sum_{k=0}^{\infty} \frac{t^{2k}}{\sqrt{(k!)}},$$ then since by first term expansion we have $f(t)\ge 1+t^2$, we get that $f(t) \rightarrow \infty$ for $t \rightarrow \infty.$ ...
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0answers
27 views

Algebraic or combinatorial proof that $(\sum_{n=0}^\infty {\frac{1}{m} \choose n} z^n )^m = 1+z$ as formal polynomials

I know how to prove this using analytic techniques (just by using derivatives of $(1+z)^{\frac{1}{m}}$, and basic facts about power series), but I was wondering if there's any way to prove this using ...
3
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1answer
47 views

Let $\sum a_n$ be a conditionally convergent sum of complex numbers. Can $\sum a_n z^n$ converge $\forall |z|=1$?

I'm fairly new to complex analysis, and I just thought of this problem, but I can't seem to find an easy proof, or an easy counterexample.
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2answers
88 views

Characterizations of $e^x$

I've been thinking about the following problem for a while: (AFAIK) the 'exponential function', $e^x$ can be characterized as the unique solution to the following differential equation with initial ...
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2answers
34 views

Check: Radius of Convergence of the Sum of these Complex Taylor Series

I just found the following Taylor series expansions around $z=0$ for the following functions: $\displaystyle \frac{1}{z^{2}-5z+6} = \frac{1}{(z-2)(z-3)} = \frac{-1}{(z-2)} + \frac{1}{(z-3)} = ...
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0answers
25 views

Power of a signature (sum of squares divided by number of elements)

I need to find some literature to study the theory of an exercise I am working on (it is from a course in digital image processing and pattern recognition). I have an $n\times n$ matrix, I have to ...
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2answers
26 views

Taylor series for $\frac{1}{az+b}$ centered at $z=0$ by substitution

I need to find the Taylor series centered at $z=0$ (i.e., the Maclaurin series) for $\displaystyle \frac{1}{az+b}$, where $a,b \in \mathbb{C}$ and $b \neq 0$. I thought it would be good to start out ...
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0answers
26 views

Radius of convergence $\sum_{n \ge 1} \frac{z^{n^2}}{(n-1)!}$ or $\sum_{n \ge 1} 2^n z^{n!}$ using

To study the power series $\sum_{n \ge 1} \frac{z^{2n}}{(1+2i)^n}$ what I do is to study the power series $\sum_{n \ge 1} \frac{z^{n}}{(1+2i)^n}$ obtaining the radius of convergence $R$ and then ...
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0answers
22 views

Differentiation and integration of power series

I'm learning calculus and my textbook states that: A power series can be differentiated or integrated term by term over an interval contained entirely within its interval of convergence. In ...
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1answer
18 views

Is this an incorrect error bound value?

In Step 3, they are determining the $(n+1)^{th}$ term. I think the proofreader just added 1, instead of subbing in (n+1). Is that right? I think the correct term should be ...
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4answers
69 views

What is the general term for $e^x/(1-x)$

What id the taylor series expansion for $\frac{e^x}{1-x}$? I know that the series expansion for $e^x$ is the sum of $\frac{x^n}{n!}$ from $0$ to $infty$. But how can I account for the $1- x$ in the ...
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2answers
53 views

Bernoulli Numbers and Tangent numbers.

Good evening. I am looking to see if there is a proof online to help guide me with the understanding that the Tangent Numbers, denoted $T_n$ and the Bernoulli numbers, denoted $B_n$ are related. It ...
3
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1answer
42 views

Power series of $\frac{1}{1+\frac{1}{4x}}$

Power series of $\frac{1}{1+\frac{1}{4x}}$ Now in an attempt to find this power series I used the known power series of: $\frac{1}{1+u} = 1-u+u^2-u^3+...$ Knowing this I simply substituted ...