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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6
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95 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
75 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
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0answers
87 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 ...
5
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0answers
64 views

Is $\int f=f-1\iff f(\cdot)=e^{\cdot}$ proved this way correct?

I saw this on math overflow and made me wonder, why does it work, is it rigorous, can we really factor like this, and where can we use similar tricks; Let $\int$ denote $\int_0^x$ Then solve $$\int ...
5
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0answers
136 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
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0answers
68 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
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0answers
52 views

What is $f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n$?

I want to understand the function $$f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n, \ \ \ \forall x\in\mathbb{R},$$ for any possible real $\alpha\geq0$. I know that for $\alpha$ integer, ...
5
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0answers
113 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
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0answers
196 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
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316 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
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0answers
77 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
70 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 ...
4
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0answers
166 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
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0answers
96 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
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0answers
131 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
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0answers
221 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
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0answers
175 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
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0answers
32 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
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0answers
172 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
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0answers
75 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
57 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 ...
3
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0answers
63 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
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0answers
65 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
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0answers
32 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
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0answers
74 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
117 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
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0answers
142 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
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0answers
122 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
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0answers
142 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
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0answers
49 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
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0answers
319 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
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0answers
277 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
290 views

Chebyshev Diff EQ

Find a power series solution about $x_0=0$ for the Chebyshev differential equation $$(1-x^2)y''-xy'+n^2 y=0,$$ as a function of of the integer $n$. Show that the solutions form a terminating ...
3
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0answers
228 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
146 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
43 views

Second order differential equation, power series method

Solve the differential equation $$(x+2)y''-xy'+(1-x^2)y=0 ; \quad X_0=1$$ using the power series method about the point $x_0=1$. I get to this step after deriving the derivatives of the ...
2
votes
0answers
55 views

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent. I use a root test: $\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=\lim_{n\rightarrow\infty} ...
2
votes
0answers
24 views

Find behavior near fixed point beyond linear expansion

this is my first question on math.stackexchange, I hope to have phrased it correctly! I have a differential equation $\text{$\frac{\text{d}x}{\text{d}t} = \alpha t^{-3}\frac{f'(x)}{f(x)}$ with ...
2
votes
0answers
32 views

How to compute the following sum?

How to compute the following sum? $$\sum_{k=1}^{\infty} \frac{k^{k-1} \cdot e^{-k}}{k!}$$ It is likely to be equal $1$ (there is an argumentation that goes back to random graphs). Moreover, i think ...
2
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0answers
30 views

Taylor polynomial converging pointwise but not uniformly?

Many standard examples of Taylor series $(\exp(x), \sin(x), \cos(x))$ converge uniformly, others don't converge to its original function at all, e.g. $\exp(-x^{-2})$. I couldn't think of any smooth ...
2
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0answers
15 views

Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
2
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0answers
127 views

What more can i do to this infinite sum?

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...
2
votes
0answers
31 views

Help understand part of the proof. Radius of convergence is $\frac{1}{\limsup |a_n|^{1/n}}$

Can you help me understand the highlighted parts of the proof. Thanks :) Theorem: Let $\sum{a_nz^n}$ be a power series, let r be its radius of convergence. Then $\frac{1}{r} = \limsup |a_n|^{1/n}$. ...
2
votes
0answers
52 views

Generators of an ideal in rings of power series

Please help me for solving a homework. Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that ...
2
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0answers
29 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 ...
2
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0answers
47 views

Origin of Laplace Transform

Is the Laplace transform the continuous version of the infinite power series? $$ \sum_{n=0}^\infty a_nx^n$$ becomes $$\int_0^\infty f(t)e^{-st}dt$$ I learned this by watching this video lecture: ...
2
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0answers
52 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
2
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0answers
48 views

Power series to solve differential equations?

We can use the formula $$F(x)=e^{λx} [ ρ-λμ-\dfrac{1}{2} λ^2 σ^2 ]^{-1}. (1) $$ to derive an expression for F(x) when f(x) is any integer power $x^n$. Begin by observing that for the ...
2
votes
0answers
100 views

Sum of $k$th power of first $n$ natural number (power sum)

I was working on a problem which involves computation power sum (summation of $k^{th}$ power of first natural number), can someone help me how to simplify the below equation. I can compute power sum ...
2
votes
0answers
115 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...