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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0
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3answers
81 views

Calculate the sum of this series

$$ \sum_{n=1}^\infty \frac{1}{n^2 3^n} $$ I tried to use the regular way to calculate the sum of a power series $(x=1/3)$ to solve it but in the end I get to an integral I can't calculate. Thanks
5
votes
2answers
67 views

Derivative of sum of powers

For fixed $n \geq 1$ and $p \in [0,1]$, is there a nice expression for the derivative of $\sum_{k=0}^n p^k (1-p)^{n-k}$ with respect to p?
-1
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1answer
28 views

Another question on finiding special kind of power series [on hold]

Let $\sum a_nx^n$ be a real power series with finite positive radius of convergence $R$ ; then does there exist a non-constant real sequence $\{b_n\}$ such that $\sum b_nx^n$ is convergent for at ...
4
votes
1answer
32 views

On finding special kinds of power series

Let $\sum a_n x^n$ be a real power series with finite positive radius of convergence $R$, then is it true that for every real number $s>0$ , we can find a real sequence $\{b_n\}$ (depending on $s$, ...
2
votes
3answers
61 views

Prove that series $ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $ and $ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $ have the same radius of convergence.

I want to prove that these two power series $$ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $$ and $$ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $$ have the same radius of convergence. What I've done so far is: ...
1
vote
2answers
30 views

Power Series and Taylor's series [on hold]

Express $f(x)=\ln x$ as a power series of $(x-2)$. I can only get the power series of $f(x)=\ln x$ but I cannot get the question where they say to express that function as a power series of $(x-2)$. ...
0
votes
1answer
22 views

Summation of finite power seires

Is it possible to find a close form solution for $S_1$. $S_1$ is defined as follows: $S_1=\sum_{k=b}^{\infty}\frac{x^k}{k!}$ ; Where $0<x<b<\infty$ If $b=0$ then $S_2 = e^x$. But how do we ...
11
votes
2answers
322 views

Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
0
votes
1answer
47 views

Extract $A+B+C$ from $A^{\frac{3}{2}}+B^{\frac{3}{2}}+C^{\frac{3}{2}}= R*D^{\frac{3}{2}}$

I need to find $A+B+C=?$ from $A^{\frac{3}{2}}+B^{\frac{3}{2}}+C^{\frac{3}{2}}= R*D^{\frac{3}{2}}$ I know that I can't use log for this equation. Do anyone have any ideas of how to do the ...
1
vote
6answers
75 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
votes
0answers
69 views
+50

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
vote
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
votes
1answer
19 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
votes
1answer
35 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
votes
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
votes
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
vote
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
vote
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
vote
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
vote
1answer
26 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
votes
1answer
229 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
vote
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
59 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
votes
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
vote
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
vote
2answers
61 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
132 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
vote
0answers
47 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
votes
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
votes
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
votes
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
vote
1answer
17 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
votes
4answers
320 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
votes
2answers
54 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
vote
1answer
47 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 ...
-2
votes
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 $12$?
1
vote
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
votes
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 $A[[x]] \to B[[x]]$?