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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7
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122 views

Can a Power Series tell when to stop?

The naive description of the radius of convergence of a complex power series is as the largest radius so that the ball avoids poles and branch cuts. This makes sense in a world where analytic ...
6
votes
0answers
48 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of ...
6
votes
0answers
127 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider ...
6
votes
0answers
77 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 ...
5
votes
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223 views

Formula for composition of formal power series with binomial coefficient

Let $f=\sum\limits_{n\geq 0}{f_n x^n}$ and $g=\sum\limits_{n\geq 1}{g_n x^n}$ be formal power series. The $x^n$ coefficient of $f(g)$ is $$ \sum\limits_{\mathbb{i} \in \mathcal{C}_{n}} {f_k ...
5
votes
0answers
87 views

Generating Function for Associated Stirling Numbers $b(n,k)$

I am trying to identify or find the ordinary generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted $$b(1;n,k)=b(n,k)$$ These ...
5
votes
0answers
141 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
5
votes
0answers
76 views

About $\prod{\left(1-q^n\right)^{5}}$

Is there a result about the non-vanishing of coefficients of $$\prod_{n=1}^{+\infty}{\left(1-q^n\right)^{5}}=1-5q+5q^2+10q^3-15q^4-6q^5-5q^6+25q^7+15q^8-20q^9+\cdots \text{ ?}$$ Thanks !
5
votes
0answers
142 views

“Evaluation Homomorphisms” for Formal Power Series

In the ring of formal power series $\Bbb R[[x]]$ it is easy to check by induction that $$ 1 = (1-x)(1 + x + x^2 + \cdots). $$ Does this derivation imply the same identity for those real or complex ...
5
votes
0answers
220 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
5
votes
0answers
334 views

Is there a reasonably simple, non-recursive formula for this sequence?

I have been curious about the following sequence of rational numbers for some time. I identified the numerator and denominators on the Online Encyclopedia of Integer sequences, but there was not much ...
4
votes
0answers
42 views

How to compute this series?

I am stuck in computing this series (i.e, finding a closed-form formula): $$ \sum_{i=0}^k \binom{k}{i} \frac{2r^{i+1}(1-r)^{k-i+1}p^{k-i}v^i s^k}{(1-r)p^{k-i}s^i + r v^i s^{k-i}}, $$ where $r$, $p$, ...
4
votes
0answers
39 views

Questionable Convergence of a Series

The summation is: $$ S = \sum_{k \geq 0} f(k) \int_{0}^{\pi/2} \sqrt{1-(1- \frac{f(k+1)^2}{f(k)^2})\sin^2(\theta)}d\theta $$ Now, we know that $f(k+1) < f(k)$ and as $k$ approaches infinity, ...
4
votes
0answers
80 views

How to solve this equation in $ \mathbb{C} $?

From a small simple calculation , we get the following formulas: $ \begin{cases} e^x = \Big( \displaystyle \sum_{n \geq 0} \frac{x^{3n}}{(3n)!} \Big) + \Big( \displaystyle \sum_{n \geq 0} ...
4
votes
0answers
76 views

Is there a formula for $\sum_{n=0}^{+\infty} q^{n^3}$?

When I studyied the representation of integers as sum of squares, I found that the most powerful tool is the Jacobi Triple Product, in fact this amazing identity allows us to find more useful ...
4
votes
0answers
92 views

Calculate $1^1 + 2^2 + 3^3 + … + n^n$

Is there a formula to calculate $1^1 + 2^2 + 3^3 + ... + n^n$ I searched but didn't find a formula for increasing powers
4
votes
0answers
65 views

Is there an algebraic description of the ring of analytic functions on the real projective line?

Apologies for the long question. Let $X=\mathbf P^1(\mathbf R) \subseteq \mathbf P^1(\mathbf C)$ be the real projective line. Let $\mathcal O_X$ be the sheaf of real-analytic complex-valued functions ...
4
votes
0answers
224 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
4
votes
0answers
100 views

Using formal power series to solve nasty equations

Consider a function $f:[0,\infty)\times \mathbb R\to\mathbb R$, and suppose that given some $a>0$, I would like to solve for $x\in\mathbb R$ satisfying \begin{align} f(\delta, x) = a. \end{align} ...
4
votes
0answers
148 views

Function composition and Bell polynomials

Suppose that we have the Taylor expansions : $$f(x)=\sum_{n=1}^{\infty}\frac{a_{n}}{n!}x^{n}$$ $$g(x)=\sum_{n=1}^{\infty}\frac{b_{n}}{n!}x^{n}$$ Then we have the standard result : ...
4
votes
0answers
225 views

Transformation of a Taylor series: “doubling” the derivative order

Suppose a function $f(z)$ has a convergent Taylor expansion: $$f(z)=\sum_{n=0}^{\infty} c_n \frac{z^n}{n!}$$ Are there general tools to compute $$g(z) = \sum_{n=0}^{\infty} c_{2n} ...
4
votes
0answers
181 views

What is known about the transformation of a power series in which $z^n$ is replaced with $z^{n^2}$?

Say we have the function $$G(z) = \sum_{n \geq 0} g_n z^n.$$ Is there a name for the transform T defined so that $$(T(G))(z) = \sum_{n \geq 0} g_n z^{n^2}?$$ Is there anything known about this ...
3
votes
0answers
41 views

How to show that two probability generating functions are equal?

From Grimmett's Probability and Random Processes: Let $G_a(s) := \sum_0^\infty a_is^i$ where $a = \{a_i : i \geq 0\}$ is a real sequence. Uniqueness. If $G_a(s) = G_b(s)$ for $|s| < R'$ ...
3
votes
0answers
79 views

40th derivative of a function

I would like to have some verification to see if my answer is correct. The given function is $f(x)=ln(1+x^2)$ and I need the 40th derivative at $x=0$. Here is my work: Using series one can manipulate ...
3
votes
0answers
35 views

Does a better form exists for the coefficients of this product of power series?

Let $$f(a,b,t) = \sqrt{1-at}\sqrt{1-bt}$$ We take the series for $\sqrt{1-at}$ and $\sqrt{1-bt}$ around $t = 0$ and multiply them together to find $$f(a,b,t) = \sum_{n=0}^\infty ...
3
votes
0answers
25 views

Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] ...
3
votes
0answers
28 views

Power Series : Interval of Convergence

Find the interval $I$ and radius of convergence $R$ for the given power series. $$\sum_{n=1}^\infty \frac {5^n}{n}x^{n}$$ What I got was that I used the limit as it goes to infinity I ended up with ...
3
votes
0answers
40 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
3
votes
0answers
107 views

Is there a way to exploit local redundancy in a function to speed up Monte Carlo integration?

In every Monte Carlo method I've ever seen, $f$ must be recomputed from scratch for each point that is (somehow randomly) selected to contribute to the overall integral. However, most functions have ...
3
votes
0answers
49 views

Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
3
votes
0answers
58 views

Intuitive explanation of proof of Abel's limit theorem

Assume the series $$f(x)=\sum_{n=0}^{\infty}a_n x^n$$ converges for $-r<x<r$. Abel's theorem says that if the series also converges at $x=r$ then $\lim_{x\to r-} f(x)$ exists and we have ...
3
votes
0answers
180 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
3
votes
0answers
124 views

Proof of Abel's theorem

I tried to prove: If $g(x) = \sum_{n=0}^\infty a_n x^n$ is a power series that converges at $x= R > 0$ then it converges uniformly on $[0,R]$. Please can you check my proof? Let $\varepsilon ...
3
votes
0answers
74 views

Prove that $e^{\ln{z}}=z$ from the power series

For my course in complex analysis we have to prove that the trivial relation $e^{\ln{z}}=z$. We are given the series for $\ln z$: $$f(w)=\sum_{n=0}^\infty (-1)^{n+1}\frac{w^n}{n}$$ $$\ln z = ...
3
votes
0answers
78 views

How come Stone-Weierstrass theorem does not imply that in a given interval every continuous function has a power series expansion?

Since for all continuous functions we get a polynomial sequence that uniformly converges to that function? As the degree of polynomial increases it should look like a power series expansion?
3
votes
0answers
56 views

A notion of transcendence degree for fields of formal power series?

There is an intuitive sense in which one would like to say that the field of formal Laurent series $k((z))$ over a field $k$ has "transcendence degree $1$". Of course, it doesn't have transcendence ...
3
votes
0answers
80 views

Is there a gap in Serre's proof of inverse function theorem?

On page 73 of 'Lie algebras and Lie groups', Serre proves the inverse function theorem for complete fields. I would like to have some clarification about the following point. Let $K$ be a complete ...
3
votes
0answers
150 views

Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
3
votes
0answers
204 views

Why does the following equation hold?

$\sum_1^\infty\frac{(k\theta e^{-\theta})^k}{k!}=\frac{\theta}{1-\theta}$, where $0<\theta<1$. It can be verified via simulation, but I haven't proved it. Are there any previous results on ...
3
votes
0answers
127 views

Changing the power series of sin(x), and its waves are getting bigger. How big do they get?

Motivation: I've been thinking about the transformation of power series, which takes the (power series of) $\exp(x)$ to $\sin(x)$. At first i was trying the series $\sum_{n=0}^{\infty} ...
3
votes
0answers
262 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) = ...
3
votes
0answers
50 views

Formal integration of a series of the type $-f(x-a)=\sum_{n=0}\frac{C_n}{n!}\delta^{(n)}(x-a)$

This question is inspired from an answer given to this question in the physics stackexchange, specifically the integration step going from (12) to (13). We have a distribution given as ...
3
votes
0answers
335 views

Under which hypotheses is switching between sum and integral signs legit?

Which hypotheses are needed to change the order of sum and integral signs? Concrete example: consider the expression $$ ...
3
votes
0answers
291 views

domain of convergence of a multivariable taylor series

consider the rational function : $$f(x,z)=\frac{z}{x^{z}-1}$$ $x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type ...
3
votes
0answers
629 views

How is Lagrange's inversion theorem derived?

I am interested in the complex-analysis version of deriving Lagrange's inversion theorem: If $y=f(x)$ with $f(a)=b$ and $f'(a)\neq 0$, then $$x(y)=a+\sum_{n=1}^{\infty} \left(\lim_{x\to ...
3
votes
0answers
234 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
3
votes
0answers
161 views

Holonomic ideals and D-finite power series

I would like to understand the connection between the term D-finite power series (in n variables) and the term of a holonomic module over the Weyl algebra A_n. A power series $f \in K[[x_1,...,x_n]]$ ...
2
votes
0answers
29 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 ...
2
votes
0answers
12 views

Series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$

As stated in the title, I'd like to compute the series $\sum_{d \geq 0} \sqrt{d} \cdot z^d$ where $z$ is a (small enough) complex number. More generally, for any real value $\alpha$, is there a ...
2
votes
0answers
41 views

Does convergence of power series on radius of convergence imply absolute convergence?

Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$. If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence? I ...