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).
5
votes
1answer
53 views
Radius of convergence of $\sum_{n = 0}^{\infty} (a_1^n + \dots + a_k^n)z^n$, where $|a_1| = |a_2| = \dots = |a_k| = 1$
Here's the problem: Find the radius of convergence of $f(z) = \sum_{n = 0}^{\infty} (a_1^n + \dots + a_k^n)z^n$, where $|a_1| = |a_2| = \dots = |a_k| = 1$, and $a_i \in \mathbb{C}$.
Since the series ...
0
votes
0answers
48 views
Taylor series (or equivalent at eps -> 0) of the integral over x of a function of x and eps
I have a function $f$ of two arguments, defined as
$$
f(x,\epsilon)=\epsilon\left( e^{-\frac{(x-\epsilon)^2}{2}} - e^{-\frac{x^2}{2}}\right) + \frac{1-\epsilon}{\sqrt{1+\epsilon}}\left( ...
0
votes
1answer
38 views
Proving that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$ with power series
Probably a simple question, but I wonder about the following:
To prove that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$, I use :
$$\exp(z_1+z_2) = ...
1
vote
0answers
51 views
Proof for closed form approximation of e
I am familiar with the derivation of $e$ from a power series, $e = \sum_{k=0}^{\infty} \frac{1}{k!} $ but have not found the proof for
the following representation in any textbook
$e = \lim_{x\ \to ...
7
votes
1answer
81 views
Is this generalization of an exercise in Stein true?
The following question is exercise $14$ in chapter $2$ in Stein and Shakarchi's Complex Analysis.
Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole ...
0
votes
1answer
54 views
uniform convergence of $\sum\limits_{n=0}^{\infty} \frac {(-1)^nx^{2n+1}}{(2n+1)!}$
Given the series $\displaystyle \sum_{n=0}^{\infty} \frac {(-1)^nx^{2n+1}}{(2n+1)!}$ does the series converge on $\mathbb R$? I found that the radius is $\infty$ and I know that for $\forall c\in ...
1
vote
1answer
86 views
Necessary conditions for not having roots
Suppose $f(z)=\sum_0^\infty a_n z^n$ has a radius of convergence of $R$. What are necessary conditions, in terms of $\{a_n\}$, for $f(z)=0$ not to have any roots?
Any combinations of real/complex ...
1
vote
2answers
48 views
Interval of Convergence of $\arctan(\frac{x}{\sqrt{2}})$
I am asked to find the power series of the function $f(x)=\arctan(\frac{x}{\sqrt{2}})$. I first found the derivative of this function which is: $f'(x)=\frac{\sqrt{2}}{2+x^{2}}$. Then I found the power ...
1
vote
0answers
15 views
Linearization of an expression involving some negative powers.
I have this adorable expression
$$\left(R_1+i\ \omega\ (L_1+L_2)-i\frac{1}{\omega\ C_1}\right)+\left( \frac{1}{-i\frac{1}{\omega\ C_2}}+\frac{1}{R_3+R_{Gap}+i\ \omega\ L_3} \right)^{-1},$$
with ...
-1
votes
0answers
17 views
Expanding a function with which theorem?
I am having trouble on How to expand 2/A - 2/(3(A-1)) to make it equal 2/(A+2/3)
Should I use binomial theorem?
Any help?
2
votes
2answers
30 views
A question on Power Series Radius Of convergence-need to check answer
What is the radius of convergence for the following power series I did some and want to check the answers
$$ \sum\limits_{n=1}^\infty {n^2\over 2^n}x^{n^2} $$
I solved it the following way but I am ...
3
votes
0answers
34 views
Maclaurin's Series of Quotients and Products
As we all (should) know, the Maclaurin series is a special case of the Taylor series when the Taylor series is centered around 0. This is the canonical definition of the Maclaurin series:
$$
f(x) = ...
2
votes
1answer
31 views
Uniform convergence of a power series
im new to this subject and very much appreciate your help with this Question, im not really sure about how to approach this, so if u can, please explain your steps
$$f(x) = ...
5
votes
2answers
113 views
Interval of convergence of $\sum\limits_{n\geq0} \binom{2n}{n} x^n$
We consider the power series $\displaystyle{\sum_{n\geq0} {2n \choose n} x^n}$. By Ratio Test, the radius of convergence is easily shown to be $R=\frac{1}{4}$.
For $x=\frac{1}{4}$, Stirling ...
5
votes
0answers
50 views
“Natural” interpolation between partial sums of a power series
Suppose $f(z)=\sum_{n=0}^\infty a_n z^n$ has a radius of convergence of $R$. Let the $N$-th partial sum be $f_N (z)=\sum_{n=0}^N a_n z^n$. What smooth (analytic) function interpolates between ...
2
votes
1answer
32 views
Find Taylor series expansion and convergence radius for $\int_0^x\cos(\sqrt{t}\ )dt$
i must find the the Taylor series expansion (i've been asked not necessarily calculating it directly) and the convergence radios for this function :
$$f(x) = \int_0^x \cos(\sqrt{t}\ ) \, dt$$
I am ...
0
votes
3answers
56 views
Series Convergence of $1/(1+x)$
For what $x$ does $\sum_{k = 0}^{\infty} (-1)^k x^k$ converge (to $1/(1 + x)$) or diverge? Or does it converge within an interval like $\left[-1, +1\right]$?
4
votes
2answers
60 views
Interval of convergence for $\sum_{n=1}^∞({1 \over 1}+{1 \over 2}+\cdots+{1 \over n})x^n$
What is the interval of convergence for $\sum_{n=1}^∞({1 \over 1}+{1 \over 2}+\cdots+{1 \over n})x^n$?
How do I calculate it? Sum of sum seems a bit problematic, and I'm not sure what rules apply for ...
3
votes
1answer
53 views
Multiplying two summations together exactly.
Consider the integral: $$\int_0^1 \frac{\sin(\pi x)}{1-x} dx$$ I want to do this via power series and obtain an exact solution.
In power series, I have $$\int_0^1 \left( \sum_{n=0}^{\infty} (-1)^n ...
1
vote
2answers
72 views
What does analytic at a point means?
A function that is analytic at a point is one that can be represented by a Taylor or Maclaurin series? We also say that the radius of convergence should be positive. What if it was negative? What that ...
0
votes
1answer
27 views
Power Law Probability Distribution From Observations
This is probably a very simple question:
I am trying to understand a power-law fitting technique by Aaron Clauset (http://tuvalu.santafe.edu/~aaronc/powerlaws/), but to do this I need to understand ...
5
votes
1answer
78 views
Proof that $\sum_{n=1}^{\infty} z^{1/n}$ doesn't converge
I believe I found a proof for the divergence of this sum for any value of $z$ besides 0.
We can look on the telescopic series: $$\sum_{n=1}^{\infty}z^{1/(n+1)}-z^{1/n} = \lim_{N\rightarrow \infty} ...
4
votes
1answer
107 views
Estimate the scale of $e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m}$
I would like to estimate the scale of the following series,
$$S(m,t)=e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m},$$
where $e$ is the base of ...
0
votes
0answers
36 views
Sumfunction of the series $\sum_{n=0}^{\infty}(2n-1)(2x)^{n}$ for all points in the interval of convergence.
The question is about to find the sumfunction of the series $\sum_{n=0}^{\infty}(2n-1)(2x)^{n}$ for all points in the interval of convergence.
Before I show you what I have tried, the earlier ...
0
votes
1answer
55 views
Calculate the Radius of convergence of $\sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$
I need your help:
Calculate the Radius of convergence of the following:
$$ \sum^\infty_1(x+1)^n\frac{(-2)^n+3^n}{n}$$
Im new to this subject, so I'd appreciate it if you can add explanations to ...
3
votes
2answers
79 views
Change of a variable in a generating function
Assuming I have a generating function
$$\sum_n c(m,n,k)x^n=\left(x\frac{1-x^m}{1-x~~~}\right)^k$$
(mentioned in this answer where $c$ represents the number of compositions of $n$ to $k$ parts of ...
8
votes
1answer
113 views
How to calculate the integral of $x^x$ between $0$ and $1$ using series? [duplicate]
How to calculate $\int_0^1 x^x\,dx$ using series? I read from a book that
$$\int_0^1 x^x\,dx = 1-\frac{1}{2^2}+\frac{1}{3^3}+\dots+(-1)^n\frac{1}{(n+1)^{n+1}}+\cdots$$ but I can't prove it. Thanks in ...
1
vote
2answers
60 views
Which sets of positive rationals are closed under addition?
This question evolved because I was interested in generalizing power series
so the exponents were rational numbers instead of integers,
i.e., $\sum_{i=1}^{\infty} a_n x^{r_n}$,
with the $a_i$ real and ...
1
vote
3answers
61 views
What's the formula for $\sum_{n=0}^{\infty}\left ( an+b \right )x^{n}$=?
Use the two formulas
$\sum_{n=0}^{\infty}=\frac{1}{1-x}$ and $\sum_{n=0}^{\infty}\left ( n+1 \right )x^{n}=\frac{1}{(1-x)^{2}}$ to find a formula for this $\sum_{n=0}^{\infty}\left ( an+b \right ...
1
vote
1answer
29 views
If I have a holomorphic function on the unit disc, do I know anything about the radius of convergence of its series expansion about zero?
I'm looking at a proof that assumes only that
$f : \mathbb{D} \rightarrow \mathbb{D}$ is holomorphic with $f(0) = 0$
The first step in the proof is to "expand $f$ in a power series centered at $0$ ...
1
vote
1answer
36 views
Radius of convergence of power series (complex)
I don't know if my reasoning is right on this exercise:
If the power series $\sum a_n z^n$ has radius of convergence $R$, which is the radius of convergence of the series $\sum a_n^2 z^n$ and $\sum ...
2
votes
1answer
32 views
What happens outside radius of convergence
A real power series $\sum_{n=0}^\infty a_n z^n$ has radius of convergence $R$. I am able to prove that for any real number $r>R$, the sequence $|a_n|r^n$ must be unbounded. Must it also tend to ...
2
votes
2answers
53 views
Essential singularities of $\frac 1{e^z-1}$
How do I show that $\frac 1{e^z-1}$ has essential singularities (instead of say, poles) at $z=2n\pi i(n\in \mathbb Z)$?
I can't figure out how to show that the function does not go to infinity near ...
-1
votes
0answers
55 views
Is the radius of convergence of the power series for $\dfrac{1}{1-z}$ centered at $0$ equal to $1$?
The radius of convergence of the power series of the function $f(z)=\dfrac{1}{1-z}$ with center $z_0=0$ is equal to $1$. Is this true or false?
2
votes
3answers
120 views
What is the answer to this limit
what is the limit value of the power series:
$$ \lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \frac{x^k}{k^{k-m}}$$
where $m>1$.
1
vote
1answer
27 views
Finding the values of $z$ s.t. $\sum_{n=0}^{\infty} \left( \frac{z}{1+z} \right)^n$ is convergent
I manipulated the series to $\sum_{n=0}^{\infty} \left( \frac{1}{1-(-1/z)} \right)^n$, which converges for $|-1/z|<1$ by geometric series. Then solving for $z$, I obtained $z>(1/\bar{z})$.
Is ...
5
votes
0answers
75 views
Complex differentiable but not analytic on circle of convergence
I'm trying to get a better handle on behavior of complex power series on the boundary of their maximal disk of convergence.
I'm reading Bak-Newman's Complex Analysis, Chapter 18.1.
A regular point ...
10
votes
4answers
179 views
Taylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?
For $α ∈ ℝ$ the function $g_α \colon B_1(0) → ℝ, x ↦ (1+x)^α$ is $C^∞$ and $g_α^{(n)}(x) = n! \tbinom{α}{n}(1+x)^{α-n}$, where $\tbinom{α}{n} = \frac{α(α-1)\cdots(α-n+1)}{n!}$ is the generalized ...
3
votes
3answers
202 views
Cauchy product on exponential-looking power series
Original posting by dioxen here: Double summation including power and factorial
I am finding some trouble in computing the following sum:
$$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac ...
6
votes
2answers
76 views
Prove that $e^a e^b = e^{a+b}$
I've read the argument in Rudin, but I think I need a little clarification
\begin{align}
e^a e^b &= \sum_{k=0}^{\infty} \frac{a^k}{k!} \sum_{m=0}^{\infty} \frac{b^m}{m!}\\ &= ...
1
vote
1answer
32 views
Absolute convergance of function series
The question is for which values of $x\in \mathbb R$, the following series absolute/conditionally converge: $$\sum_{n=1}^{\infty}\frac{x^n}{(1+x)(1+x^2)...(1+x^n)}$$ I have no idea how to solev it ...
5
votes
2answers
175 views
The value of a limit of a power series: $\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$
What is the answer to the following limit of a power series?
$$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$
2
votes
1answer
56 views
Finding convergence of the next function: $f(x)=\sum_{n=1}^\infty\frac{(\ln n)^3}{n}x^n$
How can i find whether the next function converges: $f(x)=\sum_{n=1}^\infty\frac{(\ln n)^3}{n}x^n$?
I thought about this question for quite a while, What's the trick?
1
vote
3answers
49 views
Which one is the correct series expansion?
Is
$$p^{n+1} = p^0+p^1+ \dots + p^n$$
or
$$p^{n+1} = p^0\times p^1\times \dots \times p^n\text{ ?}$$
I am confused.
please explain the correct one.
1
vote
1answer
90 views
Am I right or is Wolfram right?
Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies
$$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$
Determine a recurrence relation that ${a_n}$ satisfies.
I ...
1
vote
1answer
55 views
Writing a sum as a fraction
Express
$$\sum^{20}_{i=2}f(x)^i$$where $$f(x)=\sum_{i\geq 1}2^{i-1}x^{3i}$$ as a fraction of polynomials $p(x)/q(x)$ and simplify as much as possible.
Hmm. How to do it? Wolfram is really stupid on ...
-1
votes
0answers
37 views
Solving $m$ in $m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}}$ from $n$ and $x$
How should one proceed in order to solve $m$, where $x$ is an integer
$$m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}} $$
from $n$ and $x$ in an unconditional form, such as, for ...
2
votes
1answer
42 views
Total area of squares.
We have a square whose length is $1$ unit. Every time we rotate by $\theta$ and scale the square such as you see in the image. Does the total area of squares converge if $\theta $ goes to $0$?
2
votes
4answers
84 views
How to use “results from partial fractions”?
Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies
$$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$
The denominator can be factored into $(1-2x)(1+x)^2$. Using ...
1
vote
2answers
44 views
Using mathematical induction to prove an identity related to combinatorics
Using Mathematical induction on $k$, prove that for any integer $k\geq 1$,
$$(1-x)^{-k}=\sum_{n\geq 0}\binom{n+k-1}{k-1}x^n$$
How should I proceed? The tutorial teacher attempted this question and ...
