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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6answers
54 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...
2
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0answers
29 views

Let $f(x)$ be defined over all rationals $x$ in $[0,1]$ and let $F(n) = \sum_{i=1}^n f(\frac in)$

also define $$F^*(n) = \sum_{i=1\,\,(i,n)=1}^n f(\frac in)$$ then prove that $$F^* = \mu * F$$ where $\mu$ is the Möebius function and the $*$ means the Dirichlet convolution. I tried the Bell series ...
1
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0answers
30 views

To construct a power series such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

Let $\sum_{n=0}^{\infty} a_n x^n$ is a power series with radius of convergence $R(>0)$. To construct a power series $\sum_{n=0}^{\infty} b_n x^n$, other than $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
0
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1answer
18 views

Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...
2
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1answer
30 views

Harmonic Generating Function

I have noticed an interesting generating function involving Harmonic Numbers. $$\sum_{n=1}^{\infty}H_nx^n=\frac{\ln(1-x)}{x-1}$$ But, I have not seen a generating function involving second-order ...
0
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0answers
21 views

Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and approximate $R_n < \frac{1}{10000}$

I am tasked with the following: Find a Maclaurin series representation for $f(x)=3e^{-x^2/2}$ and use the power series to approximate $\displaystyle \int_{0}^{0.5}3e^{-x^2/2}$ with error ...
3
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5answers
120 views

Power serie of $f'/f$

It seems that I'm [censored] blind in searching the power series expansion of $$f(x):=\frac{2x-2}{x^2-2x+4}$$ in $x=0$. I've tried a lot, e.g., partiell fraction decomposition, or regarding ...
1
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1answer
31 views

Taylor Approximation of $\cos(0.02)$

Use a Maclaurin $(a=0)$ polynomial for $\cos{(x)}$ with $3$ nonzero terms to approximate $\cos{(0.02)}$. Also, use the Taylor Remainder Theorem to find a bound on the error $\left(\displaystyle ...
1
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1answer
29 views

Asymptotic Expansion of $\ f(x)=\frac{\log(x)}{\frac{\log(x)}{2\alpha}-\log(\log(x))}$

I'm looking for the asymptotic expansion as $\ x \rightarrow \infty$ for $\ f(x)$ for small $\alpha$. Ideally, I'd like to get the asymptotic expansion for all orders. How would I go about doing this? ...
1
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0answers
32 views

How can we prove $e^{x+y}=e^{x}e^{y}$ by the power series form of exponential function? [duplicate]

How can we prove $e^{x+y}=e^{x}e^{y}$ by the power series $$e^{x}=\sum_{k=0}^{\infty}\dfrac{x^{k}}{k!}\,\,\,?$$ Is there any simple method?
1
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1answer
25 views

Bernoulli-like generating function

What are the coefficients of the series for: $$\frac x{e^x+1}$$ It looks similar to the Bernoulli generating function, but the $+$ sign is throwing me off. I already found the series for its ...
2
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1answer
227 views

Partial sum of exponential series strictly increases after certain step

While trying to show that partial exponential series evaluated at two different values are strictly increasing provided that sufficient number of terms are applied I stuck at a problem. Given two ...
1
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1answer
35 views

Does any differentiable function admit an expansion $\sum^{\infty}_{n=0}a_{n}x^{n}$?

Let $f(x)$ be a differentiable function in some interval $D$. Then does that mean that we could always write f(x) in the form $\sum^{\infty}_{n=0}a_{n}x^{n}$ in that interval?
5
votes
1answer
57 views

Suppose $\sum_{k=-\infty}^{\infty}a_kz^k$ and $\sum_{-\infty}^{\infty}b_kz^k$ converge to $1/\sin(\pi z)$. Find $b_k-a_k$.

Suppose that the Laurent series $\sum_{k=-\infty}^{\infty}a_kz^k$ converges to $1/\sin(\pi z)$ when $0<|z|<1$, and suppose that the Laurent series $\sum_{k=-\infty}^{\infty}b_kz^k$ converges ...
0
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0answers
22 views

Closed form of function from a power series

Let's say we are given a power series for a function. Assuming that the power series has a closed-form representation, is there some sort of algorithim that could be used to find this closed form? And ...
1
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0answers
27 views

Convergence of a hypergeometric function

The hypergeometric function, ${}_{2}F_1(a,b,c;z)$ can be written in terms of a power series in $z$ as follows, $${}_{2}F_1(a,b,c;z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} ...
1
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2answers
60 views

Approximation of integration

I want to estimate the integral $\int_0^{1/2}\ln(1+ \frac{x^2}{4})$ with error at most $10^{-4}$. Any help will be appreciated. I have calculated the power series of $\ln(1+ \frac{x^2}{4})$ which ...
5
votes
0answers
131 views

Apartness of reals and algorithm exctraction

I am trying to wrap my head around the notion of apartness in constructive mathematics and it turns out I lack understanding miserably. I would like to use as elementary notions as possible, in the ...
1
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0answers
46 views

Expressing a function as a power series

How can I represent $$(t^2+1)^{-1/2}$$ as a power series? What I know is $$\frac1{1-t} = \sum_{n=0}^\infty t^n$$ $$-1<t<1$$ Additional: I encountered this problem when solving Legendre ...
2
votes
1answer
27 views

Geometric series not about the origin

Find a simple expression for the power series: $$\sum_{n = 1}^\infty n(z-1)^{n-1}$$ My question is can i treat this as a geo series and end up with this result: $$\frac{d}{dz} \frac{1}{1 - (z-1)} ...
0
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0answers
45 views

Question on how to manipulate terms in this expression

sorry for the vague title, i dont know how else to express what i mean with this question. But what i need to do is find out which terms on the RHS of the expression are constants. It is clear that it ...
0
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0answers
50 views

Finding closed form for this summation

recently i have beeen asking alot of questions about summations, But this one is actually quite interesting: $$ \sum_{j=k}^n j! 2^{k-2j} \left({2j-k-1 \choose j-1} - {2j-k-1 \choose j}\right){n \brack ...
0
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1answer
37 views

Expanding a function into a series

I am trying to follow a proof in QFT notes, however I am unable to follow this step - it's basically Laurent/Taylor expansion but I have very little experience with it. It's claimed that: ...
1
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1answer
15 views

Translations AND dilations of infinite series

Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ...
7
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4answers
316 views

Why do we say “radius” of convergence?

In an intuitive sense, I have never understood why a power series centered on $c$ cannot converge for some interval like $(c-3,c+2]$. Also, I have had a few professors casually mention that a series ...
2
votes
0answers
46 views

Finding a closed form for this summation

I have been trying to derive a few identities using some bell polynomials and a technique i have come up with and i came across this summation: $$ \rho(n,k) = \sum_{j=0}^k {k \choose j} {\frac{-j}{2} ...
0
votes
0answers
22 views

Perturbation theory and variable exchange of poisson-boltmann equation in spherical coordinates

I'm trying to understand this article. I think he has missing terms in his equations, and I can't understand how he derived equations 8-10. The math should be straight forward, and this make ...
2
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2answers
53 views

Simplifying a Triple Summation

I have the summation: $$ \sum_{c=1}^{n-1} \sum_{k=c}^n \sum_j \frac{\rho(n,k)}{j!(k-c-j)!(c-j)!} $$ Where the sum $j$ goes from $0$ to $k-c$ if $k-c \leq c$, but if $k-c \geq c$ then the sum goes from ...
2
votes
4answers
61 views

What is the correct radius of convergence for $\ln(1+x)$?

My text tells me this: And, Wolfram tells me this: Now, I'm not certain what to believe, but I believe I'm not certain because I'm not certain if Wolfram is using the logarithm with base $10$. ...
0
votes
0answers
20 views

convergence of a product of sums of triangular arrays

Suppose the following triangular array is given: $((a_{k,N})_{1 \leq k \leq N})_{N \geq 1}$ such that $$A_N := \left( \sum_{k=1}^N a_{k,N}^2 \right)^{\frac{1}{2}} \to \infty,$$ $$\max_{1 \leq k \leq ...
1
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1answer
46 views

Trying to understand a power series example from Advanced Calculus by Taylor

Example 2 from 21.1 in the book, Find an expansion in powers of $x$ of the function $$ f(x) = \int_{0}^{1} \frac{1-e^{-tx}}{t}dt $$ and use it to calculate $f(1/2)$ approximately. I ...
0
votes
2answers
49 views

Convergence of a product series with one $1/k$ factor

Let $\left( a_n \right)_n$ be a sequence such that $a_n < 1$, $a_n \rightarrow 0$. Prove or disprove (with a counter-example) that $$ \sum_{n=1}^{\infty} \frac{a_n}{n} < \infty.$$ Comments. If ...
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0answers
28 views

How do I prove that $f(s)=\sum a_{n}{n}^{− s}$ converge for $Re(s)>0$if the partial sum of $a_{n}$ are bounded?? [duplicate]

let $f(s)$ be a power series defined as follow :$$f(s)=\sum a_{n}{n}^{− s}$$ Assume the partial sum of $a_{n}$ are bounded .My question here is : How do I prove that $f(s)$ converge for ...
0
votes
1answer
20 views

Radius of convergence of a power series (little question about power + constant)

My power series are: $$\sum _{n=1}^{\infty }\:\frac{x^{3n+1}}{\left(1+\frac{1}{n}\right)^{n^2}}$$ So its isnt difficult if it was written without the $+1$ in the power: $$\sum _{n=1}^{\infty ...
0
votes
1answer
35 views

Find power series solution of ${x^{2}y''-xy'+py=0}$ about x=1, p is a constant.

The recurrence relation I ended up getting doesn't match with the final answer. Did a couple of revisions of the thing but nothing changes. I get the recurrence relation as ...
0
votes
2answers
27 views

Radius of convergence of series $\sum^\infty_{n=0} 3^{-n} (2 \pi)^{-n} (\arctan n)^n x^n$

Is it correct that the convergence radius of the series $\sum^\infty_{n=0} 3^{-n} (2 \pi)^{-n} (\arctan n)^n x^n$ equals $12$?
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0answers
42 views

Transfering to function when power of the x in power series is odd

Just a little question. When i have even power, it is obvious, for example: $$\sum _{n=0}^{\infty }\:x^{2n}\:=\:\frac{1}{1-x^2}$$ Is it correct to say that (when the power is odd): $$\sum ...
0
votes
1answer
53 views

Power series representation of gamma function?

I am looking for a power-series expression of the form $\Gamma(z)=b+\sum_{k=0}^\infty a_kz^k$ where the $a_k$ can be calculated as some function of k.
0
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1answer
63 views

what is the radius of convergence of power series $\frac{z^2}{z}$?

I have a power series and am being asked to find its radius of convergence, but its structure of type $$\sum\frac{z^{2n}}{z^n}$$ is confusing me. How do I calculate radius of convergence of this power ...
3
votes
3answers
49 views

Expanding $\frac{2x^2}{1+x^3}$ to series

So I was doing some series expansion problems and stumbled upon this one ( the problem is from Pauls Online Notes ) $$f(x) = \frac{2x^2}{1+x^3}$$ The actual solution to this problem uses a ...
3
votes
2answers
38 views

Flat extension of noetherian rings and formal power series

Let $A \to B$ be a flat homomorphism of Noetherian rings. Is it true that it induces a flat homomorphism of formal power series $A[[x]] \to B[[x]]$?
1
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3answers
42 views

Differentiating the exponent power series

We know that $$ e^x = \sum\limits_{n=0}^{\infty}\frac{x^n}{n!} $$ We know that the series is uniformly convergent everywhere, and therefore we can differentiate term by term, i.e $$ ...
1
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1answer
37 views

Find the residue of $e^{\frac{1}{z^2-1}}\sin(\pi z)$ at $z=1$

I'm dealing with the following problem (from an old qualifying exam): Let $\gamma$ be a closed curve in the right half-plane that has index $N$ with respect to the point 1. Find $$ ...
0
votes
2answers
29 views

Excel's EXP function compared to a series expansion

I am comparing the results of a series expansion of $e^x$ to Excel's $\mathop{EXP}(x)$ function. Should I expect them to be the same? Excel's gives $\mathop{EXP}(10) = 22026.4657948067$. However, ...
3
votes
0answers
48 views

How to find the power series of the inverse of a function?

Question: I wish to find the inverse of the following function: $f(x)=\frac{1}{2}\left(\arctan(x) + \ln\left(\sqrt{\frac{1+x}{1-x}}\right)\right)$ This is the equation for a radial null geodesic in a ...
3
votes
1answer
97 views

Limit of given expression

Let $\sum a_k=s$. I want to show that $$\lim\limits_{x\to 1^-}(1-x)\sum\limits_{k=1}^{\infty}\frac{ka_kx^k}{1-x^k}=s$$ where $x\in(0,1)$. Thanks for your helps.
3
votes
1answer
49 views

converging power series over $p$-adic integers is a UFD

Denote by $|\cdot |$ the $p$-adic norm, and let $$\mathbb Z_p \{z\}=\left\{\sum_{n=0}^\infty a_nz^n;\ a_n\in \mathbb Z_p ,\ |a_n|\underset {n\rightarrow \infty} {\longrightarrow 0} \right\}$$ the ...
1
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1answer
55 views

How to manipulate the bound on the summation

$$ B_n^{f^2}(x) = \sum_{k=1}^n\sum_{j=0}^{n-k} 2^{k-j} {j+k \choose j} \frac{d^j}{df^j}[f^k] B_{n,j+k}^f(x) $$ I am looking to have the bounds switched, can someone show me exactly how this is done? ...
2
votes
2answers
49 views

Why is the integral starts from $0$?

Consider $$f(x) = \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1}$$ It's a power series with a radius, $R=1$. at $x=1$ it converges. Hence, by Abel's thorem: $$\lim_{x\to 1^-} f(x) = ...
1
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0answers
26 views

Sum of gamma-ish power series

I'm wondering if there is a nice closed-form expression for the sum $$ \sum_{n=0}^{\infty} n^{-\alpha} x^n, \quad \alpha \in (1,2), \; x \in (0,1) $$ This is a power series with coefficients $a_n = ...