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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2
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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 ...
1
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2answers
17 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 ...
1
vote
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 ...
1
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2answers
50 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 ...
0
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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 ...
0
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0answers
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 ...
0
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1answer
30 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 ...
5
votes
2answers
56 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 $u=\arctan{(x)}$ ...
0
votes
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 ...
0
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1answer
34 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 $z=-1$?
0
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2answers
79 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 ...
0
votes
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 ...
0
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1answer
16 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 ...
0
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1answer
34 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 ...
3
votes
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
votes
1answer
24 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} ...
2
votes
3answers
46 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 ...
0
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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 ...
3
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2answers
40 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 ...
0
votes
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 $x$?
0
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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 ...
4
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7answers
547 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: $x$, $x$, $-x$, $-x$, $x$, ...
1
vote
1answer
53 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 $t \rightarrow \infty.$ ...
0
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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
votes
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.
5
votes
2answers
86 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
33 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)} = ...
0
votes
0answers
22 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 ...
0
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2answers
25 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 ...
0
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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 ...
2
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0answers
18 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 ...
1
vote
4answers
67 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 ...
1
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2answers
50 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
votes
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 ...
2
votes
4answers
57 views

Compute $\frac{1^2 t}{1!}+\frac{2^2 t^2}{3!}+\frac{3^2 t^3}{5!}+\frac{4^2 t^4}{7!}+\ldots+\frac{n^2 t^n}{(2n-1)!}+\ldots$

I have to compute $$\frac{1^2 t}{1!}+\frac{2^2 t^2}{3!}+\frac{3^2 t^3}{5!}+\frac{4^2 t^4}{7!}+\ldots+\frac{n^2 t^n}{(2n-1)!}+\ldots$$ I know that $\sinh t$ can be represented as a series, but for that ...
0
votes
1answer
23 views

How do I work out the validity for a Maclaurin (power) series?

I cannot find the answer to this anywhere so I have decided to make a question. Given a Maclaurin series for a function, how can I quickly work out what the validity is for it? For example, ...
0
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1answer
29 views

Power series $\sum_{n=0}^\infty \frac{2n+3}{(2n)!}t^{2n}$

$\begin{align*} \sum_{n=0}^\infty \frac{2n+3}{(2n)!}t^{2n}&= \sum_{n=0}^\infty \frac{2n}{(2n)!}t^{2n}+ 3\sum_{n=0}^\infty \frac{t^{2n}}{(2n)!}=\left\{\begin{array}{c} 2n=k\\ n=0\Rightarrow k=0\\ ...
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votes
0answers
10 views

Struggling with Frobenius Solutions

$x^2y''+5xy'+(x+4)y=0$ where $y = \sum_0^\infty c_n x^{n+r}$ a - prove $x=0$ is a regular singular point (done) b - find the r's (done) c - find the solution (stuck) also, I know the r's are both ...
1
vote
1answer
37 views

Finding series solution about zero

$y''+x^2y'+4y=1-x^2$ To find a power series, one substitutes in $y= \sum_0^\infty a_nx^n$. So after substitution, I've gotten $\sum_0^\infty (n+1)(n+2)a_{n+2}x^n + \sum_1^\infty (n-1)a_{n-1}x^n + ...
2
votes
2answers
30 views

Interesting power series for $y'+y=\frac1x$

I had the differential equation $y'+y=\frac1x$, which I solved for $y$ as a power series: $$y=\frac1x\sum_{n=0}^{\infty}\frac{n!}{x^n}$$ Which was a power series at $\infty$, so it doesn't really ...
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2answers
48 views

Why does the taylor series of $\frac {1}{\ln x}$ have a non-infinite radius of convergence?

Shouldn't the taylor series of a function be equal to that function for any input value? Why does this not work for the taylor series of $\frac {1}{\ln x}$ when $|x| \gt 1$? Edit: I do mean the ...
0
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2answers
35 views

When taking derivatives of power series, why do we shift the index up?

For example, if the series starts at n=0, and we take the derivative, the index usually then starts at n=1. This increases as we continue taking derivatives, but why do we need to do this? I get ...
0
votes
1answer
29 views

From known power series deduce the power series expansion of $ln(5-x)$ and infer the general term and radius of convergence

From known power series deduce the power series expansion of $ln(5-x)$ and infer the general term and radius of convergence. Now I said: ...
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0answers
22 views

a function with many branch points : the radius of convergence of its Taylor series

How can I be convinced that if a (locally holomorphic) function $f(z)$ has many branch points, say at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$, and all of the weirdest type, then the radius of ...
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0answers
12 views

Verification request- radius of convergence of two power series?

I need to find the region of convergence of : $\Sigma \frac{(n!)^2}{(2n)!}(x-2)^n$ $\Sigma \frac{x^{3n+1}}{(1+\frac{1}{n})^{n^2}}$ In 1- the series converges for $-2<x<6$, while in 2, the ...
0
votes
1answer
25 views

Expanding $1/z$ about $z=-1$ using Taylor series vs Power Series

I need to expand $1/z$ about $z_0=-1$. I decided to do it using both methods, which don't agree. Using Taylor: Finding coefficients: $$f^{(n)}(z)=(-1)^n n!/z^{n+1} \Rightarrow f^{(n)}(-1)=-n!$$ ...
1
vote
1answer
34 views

Complex Taylor Series by substitution

I need to find the first few terms or so of the Taylor series centered at $z_0 = 0$ regarding these functions: a) $e^{z\sin z}$ b)$(1+z)^z = e^{z \ln (1+z)}$ c)$\cos (1 + z^3) $ d) $e^{e^z}$ ...
1
vote
1answer
36 views

Change in Interval of convergence if center of convergence changes

So I have to find a power series that is centered at $-2^{1/2}$ If I choose to use the power series expansion for $e^x$ which converges for all $x$, and change $x$ to $x + 2^{1/2}$ does the interval ...
0
votes
1answer
20 views

General form for complex limit function $\sum p(n) z^n$ where $p \in \mathbb{C} [n]$

Given a polynomial $p \in \mathbb{C} [n]$ of degree $k$, I need to show that the power series $\sum_{n=1}^{\infty} p(n) z^n$ uniformly converges in the open unit disc, and that the limit function $f$ ...