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
34 views

Find the leftmost (most significant digits) of a large exponent calculation, say $99^{99}$

I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, $99^{99} = 3.697296 \times 10^{197}$ I only need to know the digits $3697296$ Is there any ...
1
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0answers
17 views

derivative of t distribution cdf wrt degrees of freedom

Given the cdf of a t distribution as follows: $T_\nu(x)=\frac{1}{2} + x\Gamma(\frac{\nu+1}{2}) + \frac{_2F_1 ...
1
vote
1answer
42 views

Can this infinite summation be simplified?

I encountered the following infinite summation $$\sum_{k=0,k\neq m}^{\infty}\frac{x^k}{(k-m)k!},x>0,$$ can it be simplified? Thanks!
3
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2answers
52 views

Prove two identities relating to series

Show that: $$ (1). \sum_{n=1}^{\infty}\ln\big(\cos \frac{x}{2^n}\big)=\ln \frac{\sin x}{x} $$ $$ (2). \sum_{n=1}^{\infty}\frac{1}{2^n}\tan \frac{x}{2^n}=\frac{1}{x}-\cot x $$ Thank you in advance. ...
0
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4answers
41 views

Infinite Sum of 1/Polynomial

I'm trying to solve this equation: $$\sum_{k = 0}^{\infty}\dfrac{1}{(k+1)(k+3)}$$ Original image at http://i.imgur.com/wXZFxn0.png I attempted to find the sums of $\sum_0^∞\frac{1}{k+1}$ and ...
1
vote
2answers
63 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...
0
votes
1answer
26 views

Power Series Solutions And Minimum Radius of convergence [on hold]

Help with power series and minimum radius of convergence. Does the equation $$ (x^2 + 25)y'' + xy' + x^3y = 0 $$ have a power series solution $y = \sum_{n=0}^\infty c_n x^n$? If yes, ...
0
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0answers
37 views

Remainder of $\ln x$ converges to $0$

I'm learning about power series and struggling to prove If $f(x)=\ln x$ prove that $R_n(f,c)(x)$ converges to $0$ where $c=1$. By some calculating I know that ...
0
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4answers
51 views

Power series for $f(x) = \frac{4}{x+2}$

Find the power series $f(x) = 4/(x+2)$ We know the geometric series: $$\sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$ $(x+2) = 1 - (-x - 1)$ So: $$\sum_{n=1}^{\infty} (-1)^{n-1}\cdot(x + 1)^{n-1} ...
0
votes
1answer
32 views

How to compute this series

I have to compute this series : $$\sum_{k=0}^\infty (k+1)x^{2k}$$ First, I have $$|x|<1$$ but then I don't know how to begin ...
0
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1answer
12 views

Get the closed form of Taylor series with Maple

Is it possible to get the closed form of Taylor series with Maple? The series command can give any given number of terms, but the question is about the closed form ...
0
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2answers
35 views

to find radius of convergence of power series.

I have a power series given as: $f(z) =1 + z+ \frac{z^2}{2^2} +\frac{z^3}{3!} + \frac{z^4}{2^4} \frac{z^2}{2^2}+ \frac{z^5}{5!}+ \ldots$ I have to find radius of convergence of above series. My ...
0
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1answer
27 views

If each $a_n >0$ and $\sum a_n$ diverges, prove that $\sum a_n x^n \to +\infty$ as $x\to1^-$.

Exericise 9.36: If each $a_n >0$ and $\sum a_n$ diverges, prove that $\sum a_n x^n \to +\infty$ as $x\to1^-$. (Assuming $\sum a_n x^n$ converges for $|x|<1$. Here's what I tried: Since ...
3
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2answers
35 views

How to solve linear, second order ODE with Frobenius method with a difficult recurrence relation?

The ODE in question is: $$4xy''+2y'+y=0$$ Shifting the power series of each term so that they are all raised to the power $(n+r)$ will yield this recurrence relation: $$a_{n+1}={a_n\over ...
0
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0answers
44 views

Fourier series for logarithm of sine.

I looked up here: Fourier series of Log sine and Log cos I have modified the question: How can I derive the coefficient $a_n, b_n$ for $\log(\sin(x))$ in the fourier series representation? Also, I ...
7
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1answer
82 views

How to derive this interesting identity for $\log(\sin(x))$ [duplicate]

I saw on SE that: $$\log(\sin x)=-\log(2)-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n} \phantom{a} (0<x<\pi)$$ This is an extremely useful identity, as it helps solve: $$\int_{0}^{\pi} ...
1
vote
1answer
14 views

Taylor expansion of a logaritmic function

A function is given as $ln (y) = ln(\alpha)-\frac{\lambda}{\gamma}ln(\delta L^{-\gamma}+(1-\delta)K^{-\gamma})$ I need to find the second order Taylor $ln(y)$ around $\gamma=0$. How can it be done ...
4
votes
1answer
63 views

Expressing “formally” $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series

I have to express $f(x)=\frac {1}{\sqrt {1-2x}}$ as a power series and give its interval of convergence. Knowing the binomial series is as follows this should be fairly easy: $$(1+x)^{\alpha}=\sum ...
2
votes
1answer
39 views

using power expansion to find limit

I am preparing for my final exam, and stuck on this question. Using power series expansion, evaluate $$\lim_{x\to 0} \frac{x\cos(x) -\sin(x)}{x^2-x\ln(1+x)}$$ I have no idea how to proceed. ...
4
votes
1answer
33 views

power series for $\int_0^x e^{-t^2}dt$

Use a known power series expansion to find the power series representation of the integral function $g(x) =\int_0^x e^{-t^2}dt$ centered at $a=0$ My approach Note that $g'(x) = e^{-x^2}$. ...
1
vote
1answer
36 views

Show that $f(z):=\sum a_n (z-z_0)^n$ is continuous whenever $z$ is in disk of convergence.

Consider a power series $\sum a_n(z-z_0)^n$, and assume it has radius of convergence $r$. Then we know that $\forall z\in(z_0 -r,z_0 +r)$, this power series converges absolutely by root test. Thus we ...
0
votes
1answer
19 views

Can every power series be representated as a taylor series?

Can every power series be represented as a Taylor series? More concrete: Given an arbitrary power series $\sum_{n=0}^\infty a_n (x-x_0)^n$, is there always a $C^\infty$-function $f$ such that ...
0
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2answers
61 views

Closed form of a series

Is there exist a closed form for the series of the form $$ \sum_{k=0}^{[n/2]}(-a)^{k}\binom{n-k}{k} $$ where $0<a\leq1$. For example, we have $$ ...
0
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1answer
37 views

Calculus II: Radius of Convergence

I have this math problem that states: In each part, write out the first four terms of the series, and then find the radius of convergence. $$(a) ...
0
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1answer
30 views

Taylor polynomials of degree n

I have this math question that states: Find the Taylor polynomials of degree $n$ approximating $ln(1+x)$ for $x$ near $0$. The $n$'s are 5, 7, and 9. $f^{(5)}(0)=24$; I got the derivative to ...
4
votes
1answer
86 views

Proper Bernoulli Function Generating Function

Consider the function $$\frac{t}{e^t - 1} = \sum_{i=0}^{\infty}\frac{B_i}{i!}t^i$$ This has been one of the famous generating functions for the bernoulli numbers. What about the function associated ...
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0answers
34 views

Question about series?

I have done the work so far, and have gotten these answers. a) $k=5$ b) $a_0= 20$ $a_1= -6.6667 $ $a_2= 4$ $a_3= -2.85714285714 $ $a_4= 2.2222$ c) $3.339$ d) $0.190$ All my answers ...
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2answers
31 views

Represent the function $f(x)=x^{0.3}$ as a Taylor series centered at $5$

Represent the function $f(x)=x^{0.3}$ as a power series $\sum_{n=0}^\infty c_n(x-5)^n$ Find the following coefficients: $c_0$, $c_1$, $c_2$, $c_3$ Here are my answers: $c_0= 5^{0.3} $ ...
0
votes
1answer
23 views

Time series closed form similar to harmonic series

I want to find the closed form for the following: $$ S(k, \alpha) = \sum_{t=T-k}^T \frac{\alpha^{-t}}{t} $$ when $\alpha \in (0,1)$. For harmonic series there is an easy way to upperbound: $$ H(k) ...
3
votes
1answer
42 views

A formal justification for this “physicism”?

I gave a presentation for a seminar class yesterday on Fourier analysis, and introduced the sawtooth function as a counterexample, for a function whose Fourier series is not termwise differentiable. ...
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0answers
20 views

Inequality involving radius of convergence of Taylor series

Let $(a_{n})_{n \in \mathbb{N}} \subset \mathbb{C}$ and $z \in \mathbb{C}$ . Let $f(z)=\sum\limits_{n \in \mathbb{N}} a_{n}z^n$ have radius of convergence $R_{0}$ and let $z_{0}$ be such that $|z_{0}| ...
0
votes
1answer
16 views

DE Power Series solution centered at $0$ but DE not defined at $0$.

I took a test in which I was asked to give a solution in terms of a power series for the equation: $$x^2y''(x)-xy'(x)+(1-x)y(x)=0,~~~~~~~x>0.$$ At first I began to work on a power series centered ...
0
votes
1answer
22 views

Maclaurin series of $f(x)=x^3\sin 2x$

I need help finding that maclaurin series for following function. $$f(x)= x^3 \sin2x$$ How do you get to the maclaurin series?
1
vote
1answer
31 views

Series Solution to Differential Equation

Given the series $$1 + \sum_{k = 1}^{\infty} \frac{\beta(\beta - 1) ... (\beta - k + 1)}{k!} x^k$$ how can I find a differential equation for which this series is a solution? I don't have any idea ...
1
vote
2answers
44 views

Question on the sum $\sum_{n=1}^{\infty}\frac{x^n}{n} = -\ln(1-x)$

$f(x) = \displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n} = x + \frac{x^2}{2} + \frac{x^3}{3} + ... = -\ln(1-x)$ for $|x| < 1$. $f'(x) = \displaystyle\sum_{n=1}^{\infty}x^{n-1} = 1 + x + x^2 + x^3 ...
1
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1answer
24 views

Uniform convergence of series and continuity of $f$

This is from Ross's Elementary Analysis Textbook: The series $(2^{-n})(x^n)$ from $n=1$ to $n= \infty$ represents a continuous function on $(-2,2)$, but the convergence isn't uniform. He points out ...
4
votes
2answers
47 views

Integral with series

How do I represent this integral $$\int_{0}^{1} \frac{10}{10+x^4} dx$$ as a series so that I can calculate with an error of less than $10^{-5}$.
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0answers
13 views

Gradient inequality in a simpler case

Let $f : \mathbb{R} \to \mathbb{R}$ be a analytic function. There exists $\theta \in (0,1/2]$, $c$, $\sigma$ such that for every $|x-a|\le \sigma$ $$ |f(x) - f(a)|^{1-\theta} \le c |f'(x)|. $$ This ...
0
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0answers
28 views

how to expand $\exp(-b x)$ ($b>0$) when $1<x<B$?

Let $b>0$, how to expand $\exp(-bx)$ when $1<x<B$? I am seeking a series expansion like the following ($n_0> 0$): $$\exp(-bx)=\sum_{n=n_0}^{\infty}\frac{a_n}{x^n} $$ EDIT: ...
2
votes
3answers
82 views

How to find the coefficient of a power in a power series??

How can I find the coefficient of $x^{80}$ in the power series $$(1+x+x^{2}+x^{3}+x^{4}+\cdots)(x^{2}+x^{4}+x^{6}+x^{8}+\cdots)(1+x^{3}+x^{5})\,?$$ Is there a general method to this?
0
votes
1answer
207 views

Why are there two series representations of the natural logarithm?

On the Wikipedia article of the natural logarithm one finds two different series representations for $\ln(x)$: $\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$ ...
0
votes
1answer
11 views

fast way to find order of a (semi) geometric series

Is there a fast way to find whether the order of the following is $O(T^2)$ or $O(T)$? I've been trying to find the exact thing by using the geometric series multiple time, but it is so lengthy and ...
0
votes
1answer
147 views

I have a question about integrating, and what to do about the constant. $\displaystyle\int\frac{1}{1-z}dz$

http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf On page 44, they conclude that $g'(z) = - \displaystyle\frac{\ln(1-z)}{z}$ by saying that it is just ...
1
vote
1answer
50 views

Power series $e^{-x^2}$

How would I create a power series of $f(x)=e^{-x^2}$ around $x_0=1$ without using a Taylor series? I need to know this for my upcoming exam so I would be really grateful to anyone who could show me ...
0
votes
2answers
31 views

Calculate $a_n$ with formal power series

I have $$A(x) = \sum_{n=0}^\infty a_n x^n$$ and $$A(x) = (1+x)/(1+7x+6x^2)$$ I need to find $a_0,a_1,a_2,a_3$. I multiply on each side and I get $$ (1+7x+6x^2) (a_0+a_1x+a_2x^2+a_3x^3+\ldots)=1+x ...
0
votes
1answer
21 views

Variables, Square roots, and exponents

Answer : $x^2$ I got $x^n$, shouldn't I be multiplying the variables in the parentheses first. Thus cancelling out the roots and left with $x$ then to the power of n? thus -> $x^n$ ? Please explain ...
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0answers
21 views

How to simplify this series $\sum_{m=0}^{\infty}s^m{n+m-1 \choose m}p^n (1-p)^m$ for m $\ge$ 0

I came across this is my textbook and I was wondering if anybody has a trick to simplify this series $\sum_{m=0}^{\infty}s^m{n+m-1 \choose m}p^n (1-p)^m$ for m $\ge$ 0 $$= ({\frac{p}{1-s(1-p)}})^n$$ ...
0
votes
1answer
29 views

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 ...
1
vote
2answers
27 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 ...
0
votes
1answer
39 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) = ...