Questions on dealing with series data and constructing power series expansions of functions.

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10
votes
1answer
288 views

Why is this infinite series wrongly computed by Mathematica?

Could you let me know if Mathematica (newer versions) is able to correctly compute this one? Sum[(-1)^(n + 1) Cos[3^n x]^3/3^n, {n, 1, Infinity}]
3
votes
2answers
216 views

Bessel derivatives

I tried to expand BesselJ[k,x] function into a Taylor series with Series command. Here both ...
0
votes
0answers
36 views

Series of square root of exponential

my problem concerns the Series command applied to Sqrt of an exponential, and it can be presented in a simplified version as ...
0
votes
1answer
68 views

Taylor series/expansion [on hold]

I want to make a second-order Taylor series expansion around point (x,y) = (1, Pi) of the function f(x,y)=x^2*Sin(xy/2) Plot inclusive
5
votes
1answer
182 views

Find exponential generating function from the first few terms

The function FindGeneratingFunction computes the ordinary generating function of a sequence, given a sufficient number of initial terms. I have two questions in ...
1
vote
1answer
132 views

Evaluate $\pi$ using While

I am given the following exercise: Evaluate $\pi$ using the formula $\pi = 4 - \sum_{n=1}^{\infty} (-1)^n \frac{4}{2n +1}$ with precision $\epsilon = 0.0001$. First I defined the function I need ...
0
votes
1answer
35 views

Plotting partial sums of Taylor Function

I need to plot the partial function: $T_f(x)=\log 2+\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{2^kk}(x-2)^k$ for $n=2$,$n=4$ and $n=6$ on the same plot over the interval $[-\frac{1}{2},4]$ and also the ...
12
votes
3answers
344 views

How can I obtain an asymptotic integral expansion at infinity?

I want to find the asymptotic expansion at $x \to \infty$ of the following function: $$ I(x) \equiv \int_0^{\pi/2} e^{-xt^3 \cos(t)} dt.$$ To do this I defined ...
0
votes
1answer
46 views

Series of integrate in exponential

Why Mathematica cannot perform this series: ...
3
votes
0answers
84 views

Discrepancy in the series expansion of $\log(z/(z-1))$ in Mathematica 5

When I calculate the series expansion of $\log\frac{z}{z-1}$ around $0$ in Mathematica 5.2, I obtain: ...
0
votes
1answer
41 views

Strange long time evaluation of Series for fractional function

my problem concerns the Series command applied to a product of a rational function times a square root. This can be exemplified in the following way ...
0
votes
0answers
33 views

Series Expansion of Differential Operators [duplicate]

We know about the possibility of series expansion of a function in Mathematica. Same as below, where a function f[a x] is expanded around zero up to 5th order. ...
4
votes
2answers
102 views

Taylor series representation as an infinite sum

I want to see the Taylor series representation for arbitrary functions, e.g. $\sin$. With the Series[] command, I can only see the first $n$ terms. Is there the ...
18
votes
4answers
418 views

Expansion of a hypergeometric function takes ages with Mathematica 9 and 10 (regression?)

Mathematica 8 (Linux version) can evaluate AbsoluteTiming[Series[Hypergeometric2F1[1, 1 - eps/2, 3 - eps, 1/2], {eps, 0, 0}]] in no time. On one of the ...
-2
votes
3answers
61 views
2
votes
0answers
32 views

Puiseux series for algebraic curves

Has anyone implemented a function in Mathematica that computes Puiseux expansions of algebraic curves? Using something like ...
1
vote
1answer
99 views

Changing default behavior of Series to give slightly different SeriesData

I am working with complicated expressions expr that contains the symbolic function f[x] which I know to have a pole at ...
0
votes
0answers
28 views

Subtracting Series

When I input the following $\sum_{n=0}^{m+1}x[n]-\sum_{n=0}^{m}x[n]$ which in InputForm is: Sum[x[n], {n, 1, 1 + m}] - Sum[x[n], {n, 1, m}] it returns ...
0
votes
1answer
39 views

How to turn arbitrary function to polinomial series in Mathematica?

Can I turn any multivariate function into polinomial series in Mathematica? Suppose I have a function Fwd[x_, α_] := x (1/Sin[Pi x/2])^α and wish to express it ...
6
votes
2answers
175 views

How to reverse irreversible function in Mathematica?

How to reverse formula $y(x)=x (\frac{1}{sin \frac{\pi x}{2}})^\alpha$ i.e. express it as $x = x(y)$ in Mathematica? I did this way ...
0
votes
0answers
27 views

Don't understand Series expansion in my scope

I have a quite complex, implicit problem where I have the following (short version): ...
0
votes
0answers
26 views

FourierSinCoefficients extremely slow

I'm just playing with Fourier decomposition of periodic functions. We had a similar example with standing waves and cosines in class, so I tried to expand a triangle wave in sines. This is what I came ...
5
votes
0answers
112 views

Contour Integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
0
votes
1answer
61 views

Series with a specified number of terms

I'm doing calculations with Series where I don't know the power of the leading order term. I would like to keep a specified number of terms, but since I don't know the leading order this is proving ...
0
votes
1answer
82 views

How to obtain Laurent series coefficients of an (almost) arbitrary function?

I have observed that Series in Mathematica assumes that the given function is smooth in the point around which one wants to perform series expansion. For instance: ...
2
votes
4answers
91 views

Extract a part of Series

If I have the output of Series, in terms of powers of my variable $x$, what is the quickest way to extract a part of the series, say for example the terms from $x^2$ to $x^5$, excluding those with ...
1
vote
1answer
47 views

Truncate a sum in a 'smart' way

Consider a series like this: $$\sum_{n=1}^{\infty} (c_n r^{2-n/2}+d_n r^{-n/2})$$ Sum[c[n]r^(2-n/2)+d[n]r^(-n/2),{n,Infinity}] I want, now, to keep only the ...
0
votes
1answer
28 views

Extract a term of `Series` output

Say I have the output of series, with all the coefficients of the different powers of my variable $x$. What is the quickest way of extracting the coefficient of the $n$-th power of $x$?
0
votes
1answer
47 views
2
votes
2answers
79 views

Series expression of a Root object

I was wondering if it is possible to get Mathematica to return a series approximation of a Root object. Example: I want a series representation of x in terms of ...
2
votes
2answers
133 views

Linearization of differential equations

I was wondering if one could define an operator such that, when I give a certain number of (differential) equations as an output, and an "equilibrium" value for each of the variables, it returns the ...
3
votes
1answer
161 views

Series approximation to integral

I would like to approximate the integral $$ \int_0^\infty dy\,\frac{1}{\sqrt{2\pi y\sigma^2}}\exp\left(-\frac{(x-y)^2}{2y\sigma^2}\right)f(y), $$ as a series expansion in the limit $\sigma\rightarrow ...
5
votes
5answers
127 views

Isolating cross terms

I have a set of expressions that contains terms like $\frac{1}{(x + a + b)(c+d)}$. I would like to simplify the denominator so that products of $a,b,c,d$ are dropped, but $x c$ and $x d$ are kept. ...
0
votes
1answer
66 views

Can't get solution to a system of recurrence equeations

I would be grateful if someone could help me with this problem. I define two series recursively, thus: ...
0
votes
0answers
56 views

Linearization of differential equation - need tips for the use of Series function

I would like to linearize a differential equation around a equilibrium position. The description of the steps that I have carried out are : Equation ...
0
votes
0answers
103 views

Alternative to Series

Here is a sample of my code: ...
8
votes
1answer
236 views

Error in infinite sum

The binary weight of the non negative integer k is defined by w[k_] := Total[IntegerDigits[k, 2]] The first values are (cf. http://oeis.org/ A000120) ...
1
vote
2answers
37 views

Imploying a sequence in a series

I would like to have the following sequence, $(a)_n$, in a series where $a_0 = 1$ and $a_n = a(a+1)\cdots (a+ n -1)$. The series I have is $$ 2\sum_{n=0}^{\infty}\frac{(1/2)_n}{n!(2n+1)^3} $$ I know ...
0
votes
1answer
34 views

Derivative inside a Series [closed]

I have a function defined by a sum, which I would like to differentiate. For example: ...
1
vote
1answer
53 views

Don't understand why my Taylor expansion results in a message and an unexpected result

Trying to do something simple: Taylor expand a generic function of t around a point t and substitute ...
5
votes
2answers
88 views

Replace expression in series expansion [duplicate]

A test case: I'm trying to replace an expression inside a series expansion: Series[f[x],{x,x0,4}] ./ (x-x0)->h but it still returns ...
15
votes
4answers
4k views

Multivariable Taylor expansion does not work as expected

The basic multivariable Taylor expansion formula around a point is as follows: $$ f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot ...
0
votes
1answer
84 views
10
votes
2answers
456 views

InverseSeries of multiple variables and multiple equations

CONTEXT Let us consider a bit of the Universe in which we draw spheres (see a high resolution image here). Astronomers have shown that the density within these spheres could be predicted quite ...
0
votes
1answer
48 views

Series expansion of large expression

I have the following equation: eq=E0[x,y,z]+E1[x,y,z]*Cos[phi]+E2[x,y,z]*Cos[phi]^2+E3[x,y,z]*Sin[phi] Now E0,E1,E2 and ...
0
votes
0answers
32 views

SumConvergence with product $\sum_1^\infty{\frac{1\times3\times5\times…\times(2n-1)}{n!}}$

SumConvergence[( Product[(2 n - 1), {n, 1, infinity}])/n!, n] $$\sum_1^\infty{\frac{1\times3\times5\times...\times(2n-1)}{n!}}$$ However this returns true but it ...
0
votes
2answers
57 views

Failed to use N[%] for a infinite series

I am trying to evaluate a infinite series: $$ 2 \sum _{l=1}^{\infty } \Re\left(n^{\frac{2 i \pi l}{\log (q)}} \Gamma \left(\alpha -\frac{2 i l \pi }{\log (q)}\right)\right) $$ With the code: ...
1
vote
2answers
77 views

Series with specific notation

I'm trying to get mathematicas series function to output a result that look like this: instead of calculating the actual "values", like doing so here: ...
2
votes
3answers
265 views

Sum the coefficient of a series

I am computing the Series expansion (Lauren series) of an integral and I want to sum up the coefficients of the series. ...