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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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
554 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: $x$, $x$, $-x$, $-x$, $x$, $x$,......
2
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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
28 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
51 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
90 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
37 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)} = \sum_{...
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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
23 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 $$\frac{1}{(4n+7)(2n+3)!}$$...
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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
54 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 $\frac{1}...
2
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4answers
60 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
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1answer
33 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
32 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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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 ...
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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 + 4\...
2
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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 ...
1
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2answers
53 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 ...
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2answers
38 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
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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: $ln(5-x)=ln(5(1-\frac{x}{5})=ln(5)+ln(1-\frac{x}{5})=ln(5)-\...
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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
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1answer
26 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
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1answer
39 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
37 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
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1answer
21 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 $f$ ...
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0answers
9 views

Transforming a polynomial sum using a series expansion (BCH codes)

In my study of BCH codes I've come across the following equation (the "key equation"): $$ \Omega(x) \equiv \Lambda(x)S(x) \mod x^{n-k} \tag{1} $$ Where the two terms on the right are defined by: $$ ...
0
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1answer
49 views

Radius of convergence of $\sum\limits_{n}c_{n}z^{n^{2}}$ given that the radius of convergence of $\sum\limits_{n}c_{n}z^{n}$ is finite and nonzero

I know that the radius of convergence of a given power series $\sum_{n=1}^{\infty}c_{n}z^{n}$ is $R$, where $0<R<\infty$. Given this information, I need to find the radius of convergence of $\...
1
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1answer
39 views

Radius of Convergence of Complex Power Series

I need to find the radius of the convergence of $\sum_{n=1}^{\infty}3^{n}z^{n^{2}}$ using the Cauchy-Hadamard formula. I'm not feeling 100% proficient at this method, however, so I'm asking 1) if what ...
1
vote
1answer
32 views

Find sum of power series. Having a small mistake.

Find the sum of the series. My answer is $-\frac{3}{4}$, but it should be $\frac{3}{4}$. Where did i make a mistake? $$ \sum_{n=1}^{\infty} \frac{n}{3^n} $$ $$ \frac{d}{dx} (\frac{1}{1-x}) = \sum_{n=...
0
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1answer
20 views

function representation of power series

What is the function representation of this power series? [Summation from n=0 to infinity of ($x^n)(n+1)!/n!$ The solution is $\frac{1}{(1-x)^-2}$ but how??? I know that $\sum_{n=0}^{\infty}(x^n)/n!...
0
votes
2answers
28 views

Abel's theorem power series

I am trying to show that if the power series $\sum (a_nx^n)$ coverges to a function f for $|x|<r$, then $$\int_{0}^{r} f(x)dx=\sum_{n=0}^{\infty} \frac{a_n}{n+1} r^{n+1}$$ provided that the series ...
0
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1answer
25 views

Finding the radius and the interval of convergence.

I usually use Ratio Test to find the radius and the interval of convergence. However, for this series, the ratio test does not work. If I use the ratio test, my answer is $|-2x+3|<1 $, $-2<|x|&...
2
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1answer
55 views

Need help creating a power series for with specific condtions

I needed to give an example of a power series that satisfies the following conditions: interval of convergence is [$e$,$\pi$) I came up with this series: $$\sum_{n}^{\infty}\left(\frac{2}{\pi-e}\right)...
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0answers
19 views

Confirmation on a function satisfying specific conditions(Power Series)

I had a question, find a function that satisfies the following conditions and I have to use Power series. F is the function. 1) Domain is all reals, 2) $F''(x) = cos(x^2)$, 3) $F'(0) = 3$, 4) $F(0) = ...
2
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1answer
18 views

Need help creating a power series with specific conditions

I needed to give an example of a power series that satisfies the following conditions: interval of convergence is [-1,1] and is conditionally convergence at both -1 and 1. Is it even possible to ...
8
votes
1answer
81 views

Trigonometric proof stuck with induction step

I am trying to prove: $$\sum_{s=0}^{\infty}\frac{1}{(sn)!}=\frac{1}{n}\sum_{r=0}^{n-1}\exp\left(\cos\left(\frac{2r\pi}{n}\right)\right)\cos\left(\sin\left(\frac{2r\pi}{n}\right)\right)$$ We know that ...
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2answers
43 views

Changing summation in a power series

I'm doing a question in my power series unit that involves adding summations together, I just started this unit so I'm not totally clear on how changing summation works, from what I understand you ...
0
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1answer
44 views

How to find an approximation of power series to correct to within $10^{-7}$ as faster?

If I approximate a $\displaystyle\int_{0}^{0.5} \frac{1}{1+x^7} dx$ correct to within $10^{-7}$. How to find it without using a calculator? Now I can't. I usually calculate every single term, and ...
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2answers
54 views

Sums of the series $1 + (x^2) / 3! +( x^4) / 5! +\cdots$

How can I compute sum of the series ; $$1 + \frac{x^2}{3!}+\frac{x^4}{5!}+\frac{x^6}{7!}+\frac{x^8}{9!}+\cdots$$ I tried to divide it to two pieces such that $$f(x) = 1+\frac{x}{2!}+\frac{x^2}{3!}+\...
2
votes
3answers
185 views

What is $2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty$ equal to?

I came across this question while doing my homework: $$\Large 2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty=?$$ $$\small\text{OR}$$ $$\large\prod\...
0
votes
0answers
22 views

Find a power series. centered at $x=-1$

I'm trying to find a power series, centered at $x=-1$ What do I have to do for next step? $f(x)=\frac{1}{2x-3} = \frac{1}{2(x+1)-2-3} = \frac{1}{5}[\frac{1}{\frac{2}{5}(x+1)-1}]$
0
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0answers
22 views

How to prove $\sum_{n\geq 0}{\frac{\Gamma(n+2+\alpha)}{n!\Gamma(2+\alpha)}z^n}=\frac{1}{(1-z)^{2+\alpha}}$

Let $z\in \{z\in\mathbb{C}:|z|<1\}, \alpha>-1,\Gamma(s)$ is the gamma function. How to prove $\sum_{n\geq 0}{\frac{\Gamma(n+2+\alpha)}{n!\Gamma(2+\alpha)}z^n}=\frac{1}{(1-z)^{2+\alpha}}$ ? If $...
5
votes
2answers
37 views

Exponential Power Series where Powers are Prime

I am looking for information in regards to a couple particular functions: 1) $P(x)=\sum_{p\in\mathbb{P}}\frac{x^p}{p!}$ 2) $Q(x)=\sum_{p\not\in\mathbb{P}}\frac{x^p}{p!}$ (assuming $0, 1$ are ...
0
votes
0answers
15 views

For a power series, why is the value of n changed?

I tried to find the power series of the function. $f(x) = ln(10-x)$ During this, the value of n is changed. The first term is not zero. If n is changed, it is easy to calculate. Is there any other ...
0
votes
1answer
18 views

How toexpress $V=\frac{kq}{x-a}-\frac{kq}{x+a}$ in terms of $k,q,x,u$ in Taylor Series for the following condition?

The question calls $u=\frac{a}{x}$ and $u$ is the variable. So for Taylor Series, we express it in $f(x)=\sum^{\infty}_{k=0}\frac{f^k(0)}{k!}x^k$ However, one hint says all we need is geometric ...
0
votes
1answer
58 views

Prove that $|R(z)| \leq \frac{e-1}{(n+1)!}$ if $|z| \leq 1$ Complex Variables

Let $R(z)$ be the remainder after $n$ terms in the power series of $e^z$. That is $$R(z) = e^z - \sum_{k=1}^{n}\frac{z^k}{k!}=\sum_{k=n+1}^{\infty}\frac{z^k}{k!}$$ Prove that $|R(z)| \leq \frac{e-1}{(...