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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2answers
70 views

Exact value for a series

I would like prove this equality $$1-\frac{1}{n-1}+\frac{1}{n+1}-\frac{1}{2n-1}+\frac{1}{2n+1}-\frac{1}{3n-1}+\frac{1}{3n+1}+... =\frac{\pi}{n\tan{\frac{\pi}{n}}}$$
0
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1answer
39 views

Questions regarding a complex-analytic function

So the question is formulated as follows. Given the analytic function $z \mapsto f(z) = \dfrac{1}{\sin z} - \dfrac{\cos z}{z}$, Is $z = 0$ a pole, an essential singularity, a removable singularity, ...
3
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4answers
64 views

First four terms of the power series of $f(z) = \frac{z}{e^z-1}$?

Attempt: $$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$$ $$ e^z - 1 = \sum_{n=0}^\infty \frac{z^n}{n!} -1$$ $$ e^z - 1 = z\sum_{n=0}^\infty \frac{z^n}{(n+1)!} $$ Thus $$ \frac{z}{e^z-1} = ...
0
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2answers
65 views

How does this converge uniformly?

My teaching assistant told me that $\sum \frac{(-1)^n x^n}{n}$ converges uniformly on $[0,1]$, but I doubt that. I can only see that it is uniformly convergent on $(-a,a)$ where $0<a<1$. How ...
0
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0answers
38 views

Power series and the average of its coeffients

Assume $\{b_t\}_{t\geq 0}$ is a bounded real sequence. Let $\{T_k\}_{k\geq 1}$ be an increasing sequence of natural numbers such that $\lim_k T_k=+\infty$. Suppose $$\lim_{k\rightarrow ...
1
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2answers
148 views

Regarding the radius of convergence and its equality to a certain limit

Let $f$ be a holomorphic function on the open unit disk $\mathbb{D}$, and suppose that $f$ cannot be extended holomorphically to any open set $\Omega$ containing $\overline{\mathbb{D}}$. Let $f(z) = ...
2
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1answer
96 views

Evaluate $ \int_0^1 \sum_{k=0}^\infty (-x^4)^k dx = \int_0^1 \frac{dx}{1+x^4} $

I have read this thread and I found in some comments the above named equality. I couldn't follow the transformation, which are done to get from the left to the right side at that point in particular. ...
1
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2answers
34 views

Find relative radius of convergence for this seies

Given two series $\sum _{n=1}^{ \infty} a_nz^n$ and $\sum _{n=1}^{ \infty} b_nz^n$ who both have radius of convergence $R$, show that the radius of convergence for $\sum _{n=1}^{ \infty} c_nz^n$ is at ...
3
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1answer
89 views

$(e^x-1-x)^z$ as a power series

I want to be able to write $$(e^x-1-x)^z$$ as a power series, so since $$e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$$ I can write $e^x-1-x$ as ...
4
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3answers
137 views

the sum of $1-\frac{1}{5}+\frac{1}{9}-\frac{1}{13}+…$

I thought this was the real part of the series: $\sum_{n=0}^\infty \frac{i^n}{1+2n}$, with $i=\sqrt{-1}$. When taking the real part I am left with: $\sum_{n=0}^\infty \frac{\cos(n\pi/2)}{1+2n}$. I ...
0
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1answer
43 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
0
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1answer
40 views

The accuracy of approximating $ f(x) = x^{2/5}$ for $0.9 \le x \le 1.1$ using the cubic Taylor polynomial

For the equation $ f(x) = x^{2/5}$, $a=1$, $n=3$, $0.9 \le x \le 1.1$ I was able to approximate f by the following Taylor polynomial: $$ F_3(x) = 1 + \frac2 5 (x-1) - \frac3{25}(x-1)^2 + ...
0
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0answers
13 views

Convergence domain of Power Series(Rienhardt domian)

A complete Reinhardt domain is a convergence domain for some power series if and only if the domain is logarithmically convex. i have some idea to prove the convergence domain is logarithmically ...
2
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1answer
113 views

1-Associated Stirling Number of the Second Kind identity verification

I recently posted this in regards to Associated Stirling Numbers of the Second Kind (SNSK) and I was trying to fix my equations to find and identity, and am now looking for verification that this ...
1
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2answers
69 views

Proof that sum of power series equals exponential function?

I have found that the Sum series equal an exponential function as below, however I have not found a proof for it: $$ ze^z = \sum_{k=0}^{\infty} k \frac{z^k}{k!} $$ I have though managed to prove ...
1
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1answer
41 views

Answer difference of same series with different index

Consider these two series $$\sum_{n=1}^{\infty}3\left(\frac {1}{2}\right)^n=3$$ $$\sum_{n=0}^{\infty}3\left(\frac {1}{2}\right)^n=6$$ Everybody knows that there should be a difference, What I know ...
1
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1answer
22 views

Test validity of approximation by multiplying 2 series

How can I best explain that the approximation of these two series is valid? $$ e^{-t} \cos(2t) \approx 1-t $$ The test should be made by multiplying the series. I looked at the series and am now ...
8
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1answer
191 views

$x''= \frac{Ax+B}{Cx+D}$

Might there be a closed-form solution to the second-order differential equation below?$$x''(t)=\frac{Ax+B}{Cx+D}$$ If not, is there any way to get a power series approximation in terms of the ...
0
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0answers
16 views

Determining convergence of series of form $\sum a_n z^{n^k}$

So, title says it all. Say I have a (complex) series of the form $$\sum_{n=0}^\infty a_n z^{n^k},$$ for some $k\in \mathbf{N}$. I'm a little at loss what to do with it to determine its radius of ...
2
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1answer
38 views

Confusion about a certain series expansion

While reading some old notes on contour integration, I noticed the author uses series expansion: $$\frac{\sinh sx}{\sinh ...
2
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3answers
134 views

Matrix exponential: Formal notation for power series? Or, more?

For a square matrix $A$, I'm already used to see and use: $$\sum_{n=0}^{\infty} \frac{A^n}{n!} = \lim_{n \to \infty} \left(I + \frac{A}{n}\right)^n = e^A$$ Which means a matrix $A$ is just like some ...
0
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1answer
35 views

(complex variables) Expand $\frac{2z+3}{1+z}$ in a power series of $z-1$ and comment on its convergence

Question: Expand $\frac{2z+3}{1+z}$ in a power series of $z-1$. What can we say about its convergence? Attempt: First, notice $ \frac{2z+3}{1+z} = \frac{2z+3}{1} \frac{1}{1+z}$. Let $w = 1 -z$. Using ...
0
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0answers
44 views

Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le ...
1
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2answers
43 views

Computing a large coefficient in a power series expansion

What is the coefficient of $x^{1000}$ in the power series expansion of $$\frac{1}{(1-x)(1-x^5)(1-x^{10})(1-x^{25})}?$$ This is the number of ways to break ten dollars into pennies, nickels, dimes, ...
3
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3answers
81 views

How can we show that $ \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 $? [duplicate]

How can we prove the following? $$ \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 $$ It would be great to see multiple ways, or hints, about how this can be proven. I know this is a power series ...
0
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1answer
33 views

Use Leibniz' formula to show that the $(2n)$th derivative of $(2x^2 + 3x +1)sinx$ is $(-1)^n(2x^2+3x-8n^2+4n+1)sinx+(-1)^{n+1}(8nx+6n)cosx$ wrt $x$

If I let $f=f(x)=sinx$ and $g=g(x)=2x^2+3x+1$ and $D=$ First derivative wrt $x$, $D^2=$ Second derivative wrt $x$ and $D^n=$ $nth$ derivative wrt $x$ then, Leibniz' formula states that $\displaystyle ...
0
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0answers
35 views

Power series $\sum \left(1+ \frac{2}{n}\right)^{n^3 + n^2 +1} x^{n^2} $

It is asked to find the interval of convergence of the power series $$\sum \left(1+ \frac{2}{n}\right)^{n^3 + n^2 +1}x^{n^2}$$ But how should I write this series so it has the form $$\sum ...
0
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2answers
34 views

Find radius convergence of power series

Find the radius of convergence of the series $$\sum\limits_{n=0}^\infty 3^nz^{n!}$$ My approach is as follow: ...
0
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0answers
23 views

Uniform and absolute convergence of $\sum \frac{2^n}{n^2+1} x^{2n}$

Find the radius of converence of the series $$\sum \frac{2^n}{n^2+1} x^{2n}$$ and analyze the absolute and/or uniform convergence My attempt The convergence radius of this serie is ...
2
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0answers
52 views

Generators of an ideal in rings of power series

Please help me for solving a homework. Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that ...
1
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2answers
62 views

Taylor series of Infinitely differentiable function with nonnegative derivatives

Let $f(x)$ be a nonnegative and infinitely differentiable function on $[-a,a]$ to $\mathbb{R}$ such that $\forall x\in[-a,a]:f^{(n)}(x)\ge0$. Prove that the series: $$\sum_{i=1}^\infty ...
0
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2answers
30 views

Uniform convergence on the interval of convergence

I am with a lot of doubts about the uniform convergence of a power series. For example, consider the series $$\sum \frac{1}{n} x^n$$ It is easy to find that the radius of convergence of this series ...
1
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0answers
32 views

tricky question regarding Series, Limits and Convergence

Suppose we have 2 real power series for every $k=1,2,3,\ldots$ $$ f_k(x)=\sum_{n\in\mathcal{S}^{(+)}(k)} x^n d_n(k),\quad g_k(x)=\sum_{n\in\mathcal{S}^{(-)}(k)} x^n d_n(k), \quad d_n(k)\geq 0 $$ where ...
2
votes
1answer
45 views

Does n power of e grow much more faster than its Maclaurin polynomial? [duplicate]

I wonder how to calculate the following limit: $$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{{}n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$ In the first sight, I think it should be zero, because ...
4
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1answer
111 views

Convergence of the power series $\sum \left(\frac{n^n}{n!} x^n \right)$

Find the convergence radius of the serie $$\sum \frac{n^n}{n!}x^n $$ and analyze the absolute convergence and/or uniform. What I've done: It is easy to show that the radius of convergence of this ...
4
votes
2answers
88 views

Euler's Bernoulli number Identity help

My professor and I derived the $n$th Bernoulli number as the recursion $$B_n=\frac{-1}{n+1}\sum_{k=0}^{n-1}\binom{n+1}{k}B_k.$$ Later, in a paper I was reading, there is a similar identity attributed ...
0
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2answers
42 views

How to show $f$ is a constant zero function…

Power series $f(x)=\sum\limits_{n=0}^\infty$ $a_n x^n$ with the radius of convergence $R>0$ And the sequence $(b_k)$ satisfies $R>b_1>b_2>..., $ $\lim_{k\to\infty} b_k=0$ Need to show that ...
0
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1answer
37 views

Determining the domain of holomorphic function, the taylor series of function with its convergence's radius.

I need some help and correct my knowledge, please. Let $f(z)=(e^{z}-1)/(1+z+z^{2})$. Determine the largest domain $\Omega$ in $\mathbb{C}$ such that $f$ is holomorphic in $\Omega$. Since ...
1
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1answer
29 views

Power series coefficients

I've been trying for days now to find a closed form for the coefficients of the power series about $x=0$ of the function $$ f(x)=\exp\left(r^2\frac{x(n-2)-x^2(n-1)+x^n}{(x-1)^2}\right), $$ but I ...
0
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1answer
27 views

question involving power series for ln(x+a)

please could you help with this question. If a and b are small compared with x, show that $$ln(x+a) - lnx = \frac{a}{b}(1 + \frac{b-a}{2x})(ln(x+b) - lnx)$$ I've tried expanding ln(x+a) as a taylor ...
6
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0answers
88 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider ...
1
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2answers
29 views

Series Coefficient Convergence implies Uniform Convergence

Trying to find a reference for the following. Define the entire functions, $$f_n(x)=\sum_{k=0}^\infty a_{n,k}x^k\ \ \ \ \ \ \ \ \ \ \ f(x)=\sum_{k=0}^\infty a_kx^k.$$ If for each $k$, ...
0
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2answers
18 views

Radius convergence of a power series…

"Suppose that $\sum_{n=0}^{+\infty}a_nx^n$ has convergence radius $R$, $R>0, \text{or }R=+\infty$. Proof that the convergence radius of $\sum_{n=0}^{+\infty}na_nx^{n-1}$ is also $R$." This seems ...
1
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1answer
21 views

Finding the convergence radius of a power series

Let $\sum\limits_{n=0}^{+\infty}a_nx^n$ be a power series. Prove that If $\large \lim\limits_{n\to\infty}a_ns^n=0,s>0$, then the power series above converges absolutely for $|x|<s$. ...
0
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1answer
39 views

Prove that $\sum_{k=1}^\infty\frac{1}{16k^4 - 1} = \frac{1}{2} - \frac{\pi}{8}\coth(\frac{\pi}{2})$

I want to prove that: $$\sum_{k=1}^\infty\frac{1}{16k^4 - 1} = \frac{1}{2} - \frac{\pi}{8}\coth\left(\frac{\pi}{2}\right)$$ Using the fourier series: $$\phi(x) = \begin{cases}0 & \text{if ...
2
votes
1answer
48 views

Convergence of a crazy power series

"Let $\alpha$ be a given real number, $\alpha>0$ and $\alpha \notin \Bbb{N}$. Proof that the series $$\sum_{n=1}^{+\infty}\frac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-(n-1))}{n!}x^n$$ converges for ...
1
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2answers
46 views

Conditions for convergence of $\sum\limits_{n=1}^\infty{a^nf(n)}$

assume $a>0$, and for all $n$ we have $0 \leq f(n) \leq 1$. Is there a necessary and sufficient condition on the series $f(n)$ for which $\sum\limits_{n=1}^\infty{a^nf(n)}<\infty$ ? Thanks!
0
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1answer
31 views

Uniform convergence of the series $\sum_{n=1}^{\infty} \frac{\cos(2nt)}{4 n^2 - 1} $

I am trying to find if this series is uniformly convergent: $$\sum_{n=1}^{\infty} \frac{\cos(2nt)}{4 n^2 - 1} $$ So far I have (using the Weierstrass M-Test): $$| \frac{\cos(2nt)}{4 n^2 - 1}| \le ...
1
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1answer
49 views

Product Of Series With Increment Powers

I found this interesting aptitude question and I don't know how to solve this genre of question. Any help is welcome :) $$\prod_{n=1}^{49}n^n=1¹\cdot 2²\cdot\ldots\cdot49^{49}=?$$ Thanks.
1
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4answers
30 views

Function as a series :

Let $f(x)=\sum_{n=0}^{+\infty}\dfrac{x^n}{n!}$. Verify that $$\int_0^xf(x)dt=f(x)-1$$ This is the exercise 3 of the section $7.4$, of Guidorizzi's Calculus, Vol. 4. What I have tried: By the ratio ...