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).
4
votes
2answers
44 views
expand $ \arctan\left(\frac{3x+2}{3x-2}\right)$ into pwer series, find radius of convergence (check solution)
I would be grateful if someone could check what I've worked out:
$$ f(x)=\arctan\left(\frac{3x+2}{3x-2}\right)\implies f'(x)=\frac{1}{1+(\frac{3x+2}{3x-2})^2}\cdot \frac{3(3x-2)-3(3x+2)}{(3x-2)^2}$$
...
1
vote
3answers
62 views
Expand into power series $f(x)=\log(x+\sqrt{1+x^2})$
As in the topic, I am also supposed to find the radius of convergence.
My solution: $$\log(x+\sqrt{1+x^2})=\log \left ( x(1+\sqrt{\frac{1}{x^2}+1})\right )=\log(x)+\log(1+\sqrt{\frac{1}{x^2}+1})$$Now ...
1
vote
3answers
37 views
power series of a matrix well-defined
I am working on a seminar lecture and have found the following lemma without a proof:
Given a convergent power series $f(z)=\sum_{n=0}^\infty a_nz^n$ and a diagonalizable matrix $M$ with diagonal ...
2
votes
4answers
38 views
Find expansion around $x_0=0$ into power series and find a radius of convergence
My task is as in the topic, I've given function $$f(x)=\frac{1}{1+x+x^2+x^3}$$
My solution is following (when ...
0
votes
1answer
49 views
Series Summation 1 to infinity [duplicate]
Could someone help me with this series summation:
$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots\ ?$$
I checked the answer to be $\dfrac{\pi^2}{6}$, but couldn't figure out how.
2
votes
2answers
46 views
Analytic functions of a real variable which do not extend to an open complex neighborhood
Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
2
votes
1answer
72 views
Show that some $C^\infty$ real function is analytic
Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
2
votes
2answers
112 views
$\sum (-1)^n/n$ fails the p-series test, but passes the alternating series test?
P-Series Reference
Alternating Series Test Reference
$$ \sum_{i=0}^\infty \frac{(-1)^n}{n} $$
This alternating series fails the p-series test because the exponent of n = 1.
Yet it seems to pass ...
2
votes
3answers
120 views
Difficulties performing Laurent Series expansions to determine Residues
The following problems are from Brown and Churchill's Complex Variables, 8ed.
From §71 concerning Residues and Poles, problem #1d:
Determine the residue at $z = 0$ of the function ...
0
votes
2answers
59 views
Show that cosh(2) is between two values.
I'm reviewing for exams and this question has got me stumped:
Show that:
$3\dfrac{2}{3} \leq \cosh(2) \leq 3\dfrac{2}{3} + 0.1$
I've determined the series form of cosh(x) to be:
...
2
votes
1answer
56 views
Why do power series converge to a function symmetrically?
Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?
The selected answer to the above question says that for a a power series, the interval of convergence for the ...
0
votes
3answers
42 views
Power Series Proof w/ Binomial Coef.
Prove that, for any positive integer k,
$\sum_{n=0}^\infty {{n+k \choose k}z^n}$ = $1/(1-z)^{k+1}, |z| < 1$
2
votes
3answers
64 views
Radius of convergence of fast converging power series
Suppose $a_i\ge 0$ and $a_{n+1}+a_{n+2}+\cdots < 1/n!$. What can I say about the radius of convergence of
$$\operatorname{f}(x) = \sum_{n \ge 0} a_n x^n$$
The above condition gives that ...
2
votes
0answers
18 views
Bivariate generating functions and diagonal like recurrences
I'm trying to solve recurrences of the type
$$a(n,m) = \sum_{k=0}^{m} a(n-k,k), \qquad a(n,0)= a(0,m)=1 \qquad (A_0)$$
with the help of generating functions, but I get stuck quite early on.
If I ...
2
votes
2answers
25 views
Radius of Convergence for $\sum \frac{[1\cdot 3 \cdots (2n-1)]^2}{2^{2n}(2n)!}x^n$
I'm trying to find the radius of convergence for this series:
$$\sum \frac{[1\cdot 3 \cdots (2n-1)]^2}{2^{2n}(2n)!}x^n$$
so I have,
$$R=\lim_{n\to\infty} \frac{[1\cdot 3 \cdots ...
2
votes
2answers
133 views
Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?
I'm looking for an intuitive understanding instead of a formal proof. Thanks for the help.
1
vote
3answers
32 views
Demonstrate that $\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6}\ge 0$
How do I prove the inequality?
$$\frac{1}{e^e} - 1 + e - \frac{e^2}{2} + \frac{e^3}{6} \geq 0$$
I can see that $e^e = \sum_{k=0}^{\infty} \frac{e^k}{k!} = 1 + e + \frac{e^2}{2} + \frac{e^3}{6}+\dots ...
1
vote
2answers
35 views
expressing $\int tan^{-1}tdt$ as a power series
I have a question which asks me to state $\int_0^x\tan^{-1}(t)dt$ as a power series in $x$ and then use that result to show that $\frac{\pi}{4}-\log\sqrt{2}= 1-1/2-1/3+1/4+1/5--++\cdots$
For the ...
0
votes
2answers
46 views
Taylor series of $f(x^2)$
If you know the taylor series for $f(x)$ can you find the taylor series for $f(x^2)$ by letting $x = x^2$? The taylor series in question is $\cos(x^2)$
I know the taylor series for $\cos(x)$ is ...
3
votes
3answers
81 views
Power Series Representation of $(1+x)/(1-x)$
For the power series representation of, $f(x) = \frac{1+x}{1-x}$ which is $1 + 2 \sum_{n=1}^\infty x^n$, Where does the added $1$ in front come from? How do I get to this answer from ...
0
votes
1answer
47 views
Radius of convergence of a power series
What is the radius of convergence of :
$\sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3} z^n$
0
votes
0answers
43 views
Solutions of Chebyshev equation about $x = 1$
I need to find two solutions of the Chebyshev equation: $(1-x^2)y'' - xy' + a^2y = 0$, where a is some constant. Attached is a picture of what I have so far. I apologize if its kind of hard to read. ...
1
vote
1answer
63 views
First few coefficients of a power series
The function $f(x) =\frac{5}{1+16x^2}$ is represented by the power series $$\sum_{n=0}^{\infty} c_nx^n$$
I'm supposed to find the first few coefficients of the power series, and these are the answers ...
0
votes
1answer
90 views
definite integral approximation using taylor series
In this question I cannot understand why they integrate to get the function that the power series represents...I also don't get how they selected the uppser and lower bound on the definite integral. ...
1
vote
3answers
45 views
Coefficients of series given by generating function
How to find the coefficients of this infinite series given by generating function.$$g(x)=\sum_{n=0}^{\infty}a_nx^n=\frac{1-11x}{1-(3x^2+10x)}$$
I try to expand like Fibonacci sequences using geometric ...
0
votes
2answers
41 views
Power series coeffieients
Determine the coefficients of the power series that defines a function with the following
properties: $f''(z) = −f (z), f (0) = 1, f'(0) = 0.$
1
vote
1answer
48 views
Why do Maclaurin series approximate a function for negative domain values?
A common analogy used as an intuitive explanation for a Maclaurin series is that of a car. If you know the position, velocity, acceleration, jerk etc. of a car at time zero, you are able to predict ...
5
votes
1answer
30 views
Power series for $\cos(n\theta)$ in terms of $\sin^{2i}(\theta/2)$?
Does anybody know an expression for the weights in
$$
\cos(n\theta) = \sum_{i=0}^n c_i \sin^{2i}(\theta/2)
$$
I checked the standard sources (Abramowitz & Stegun, Gradshteyn & Rhyzik) and ...
4
votes
2answers
67 views
Finding a closed form expression for this sum [duplicate]
For non-negative $n$, find
$$
\sum_{k=0}^n \binom{2k}{k}\binom{2n-2k}{n-k}.
$$
I can't figure this out. Any ideas?
2
votes
4answers
36 views
Radius convergence of power series
How would I go about finding the radius of convergence using a limit ratio test? Can I get a hint for this one?
$\displaystyle\sum_{j=1}^\infty\frac{(jx)^j}{j!}$
0
votes
2answers
34 views
Power Series of Linear Transformations
In a course I'm taking, we're talking about polynomials of linear operators, i.e. if $E$ is a linear transformation on $V$ and $p$ is a polynomial then we can consider the linear transformation ...
1
vote
0answers
44 views
Taylor series of Fourier series of triangle wave
Odd triangle wave $\text{t}(x)$ with angles at $(2x+1)\in\mathbb{Z}$ can be represented by Fourier series:
...
1
vote
1answer
52 views
Coefficients of powers of the theta function
Let $q=\exp(2 \pi i z)$ and $$\theta(z)=\sum_{n=-\infty}^\infty q^{n^2}.$$
Now, I shall show that the powers of $\theta$ are given by
$$\theta(z)^r = \sum_{n=0}^\infty S_r(n) q^n$$
where $S_r(n)$ ...
1
vote
1answer
50 views
Taylor expansion with integral?
I have looked at a version of a Taylor expansion that has an integral- for the first time. Is this the same as the usual version of a Taylor expansion without integrals? Also, do the $\alpha's$ have ...
0
votes
1answer
50 views
Demonstrating the coefficients of the power series of $\frac{1}{1-z-z^2}$ satisfies a recurrence relation.
I have the power series $$\frac{1}{1-z-z^2} = \sum_{n=0}^{\infty} c_nz^n$$ and I'd like to show that the coefficients of this power series satisfy $c_n=c_{n-1}+c_{n-2}$. I thought the most obvious way ...
4
votes
1answer
50 views
Is there a real power series with radius of convergence 1 that converges uniformly on (−1,1)?
Is there a real power series with radius of convergence $1$ that converges uniformly on $(−1,1)$?
I am guessing the answer is yes, if we can construct a function with power series such that it ...
0
votes
0answers
65 views
What is the bound of coefficients of the series $e^t$?
I'd like to get a bound of the coefficients of the series:
$$e^t = 1 + \frac{t^1}{1!} + \frac{t^2}{2!} + \dots + \frac{t^n}{n!} + \dots$$
In other words, after $n$ terms we get a term
...
1
vote
5answers
160 views
Use Cauchy product to find a power series represenitation of $1 \over {(1-x)^3}$
Use Cauchy product to find a power series represenitation of
$$1 \over {(1-x)^3}$$ which is valid in the interval $(-1,1)$.
Is it right to use the product of $1 \over {1-x}$ and $1 \over ...
7
votes
1answer
107 views
Does a convergent power series on a closed disk always converge uniformly?
If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
2
votes
0answers
148 views
power series of arcsin(x) centered at x = 0
I am trying to prove that the Taylor expansion of $\arcsin(x) = \sum\limits_{n=0}^\infty \cfrac{(2n!)x^{2n+1}}{(2^nn!)^2(2n+1)}$.
Sorry about the notation, I'm not sure what syntax to use. S stands ...
1
vote
4answers
99 views
Absolute convergence of the series $\sum_{n=1}^{\infty} (-1)^n \ln\left(\cos \left( \frac{1}{n} \right)\right)$
This sum
$$\sum_{n=1}^{\infty} (-1)^n \ln\left(\cos \left( \frac{1}{n} \right)\right)$$
apparently converges absolutely, but I'm having trouble understanding how so.
First of all, doesn't it ...
1
vote
1answer
39 views
Power series and power series expansion
I am looking for help with a problem. Here is the question I am working on:
Consider the power series
$$\sum_{n=1}^{\infty}(-1)^n \frac{n+1}{n}x^{n}$$
(a) Determine the radius of ...
1
vote
1answer
34 views
Trying to show that the product of two power series equals 1.
I've reduced a large homework problem to the following smaller problem.
Let $P = \sum_{i=0}^\infty a_i X^i$ denote a formal power series over a field. Assume $a_0 \neq 0$, and define $Q = ...
1
vote
2answers
46 views
If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero
If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero.
How can I able to prove the above problem without ...
1
vote
1answer
72 views
Where are power series uniformly continuous?
As far as I know, $f(x)=\sum\limits_{n=0}^\infty a_n(x-x_0)^n$ is continuous on the whole convergence interval $K:=\{x\in\mathbb R:|x-x_0|<r\}$. Is there anything we could say about uniform ...
1
vote
1answer
67 views
Adding two Power series or Maclaurin sums together and their radius of convergence
Say you have two power series. One of them has ROC of 2, and the other one has an ROC of 4. If you add the two series together is the ROC ALWAYS the lesser ROC? It seems to be a trend I've noticed, ...
0
votes
0answers
40 views
Taylors Inequality to evaluate $f(x) = x\sin(x)$ when $a = 0$ and $-1\le x\le1$
Trying to calculate the error of this function when you use a Taylor expansion to degree 4.
I keep getting $.039$ when the answer in the back of the book is $.042$.
I take the fifth derivative of ...
1
vote
0answers
53 views
Expansion in powers
Let $n=2k, k \in Z_+$. Let
$$P_k\left(\frac{t}{\sqrt n}\right)=n!\sum_{\begin{smallmatrix} n_1+\ldots+n_k=n \\ ...
3
votes
3answers
55 views
Is the degree function well behaved over power series?
For non-zero formal polynomials $x$ and $y$ it holds that $\deg(xy)=\deg x + \deg y$. Allowing for infinite degrees, does this formula hold for arbitrary non-zero power series? And is there a ...
1
vote
1answer
38 views
Notation for the coefficient of the $i$th term of formal power series.
What notation is standard for the coefficient of $X^i$ in a formal power series $P$? I was thinking of $X^i \cdot P$, by analogy with the dot product.
