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

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0
votes
1answer
52 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: ...
5
votes
5answers
108 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. ...
-1
votes
0answers
39 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 ...
3
votes
1answer
110 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 ...
7
votes
1answer
206 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
35 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
27 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
41 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
77 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 ...
0
votes
0answers
96 views

Alternative to Series

Here is a sample of my code: ...
0
votes
1answer
73 views

Series expand root of quartic polynomial containing many real parameters and FullSimplify return `Indeterminate`

Using Mathematica I can get the Eigenvalues of the following matrix (using Quartics->True): ...
0
votes
1answer
42 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
1answer
51 views

Replacement of terms/Pattern matching involving products of derivatives of a function

In delving into Ramanujan summation, I'm trying to get a hold of the relations of the form $$\sum_{n=0}^\infty f(n)=\dfrac{h\frac{d}{dx}}{{\mathrm{e}^{h\frac{d}{dx}}}-1}\int_{0}^\infty ...
0
votes
0answers
31 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
53 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
225 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. ...
1
vote
1answer
35 views

Series power with unusual denominator

I have a series whose terms are: (x^n)/(n+a) where n is a positive integer, x a real number that is greater than 0 and smaller than 1, and a is a real number smaller than 1. It is easy to proof that ...
3
votes
1answer
36 views

Defining the value of variable after expansion

I have an initial equation defined as: x = Subscript[a, 0] + (1 - r^2)/Sqrt[1 + r^2 - 2*r*Cos[\[Theta]]]; I want to taylor expand this after subbing in: ...
3
votes
1answer
87 views

List interpolation

Hi mathematica people! So i am looking for the best way to interpolate a function given a list of its values. I have an iterative algorithm which needs high precision otherwise the numerical noise is ...
0
votes
1answer
71 views

How to obtain Padé approximant of $\log(x)$? [closed]

Having read this answer on Math.SE, I wanted to try seeing how Padé approximations converge to $\log(x)$. But my first attempt after reading the documentation failed: ...
0
votes
1answer
56 views

Single formula for this sequence [closed]

Consider the following sequence. 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8 We can express this sequence by (an) where an=n/2 when n is even and an=0 when n is odd. Find a single formula for an ...
4
votes
3answers
193 views

Polynomial with alternating sign coefficients from the odd degree terms of a Truncated Power Series

I'm trying to make a function that will take any truncated power series I give it, strip away all the even degree terms, then change the sign of the coefficients of every other term. So that we have a ...
2
votes
1answer
86 views

Evaluate function defined by DifferentialRoot

I have the following sequence of rationals that I want to find the generating function of: ...
4
votes
1answer
137 views

How do I find a series solution for $$e^{-\frac{1}{2}f'(x)} cosh( f(x) ) = ax + b$$

I am trying to approximate a function $f(x)$ satisfying a relation between $f(x)$ and its first derivative. How do I find a series solution for $$e^{-\frac{1}{2}f'(x)} cosh( f(x) ) = ax + ...
1
vote
1answer
67 views

Series expansion of InterpolatingFunction obtained from NDSolve

I am trying to obtain a series expansion of the numerical solution of a differential equation. I encounter difficulties going beyond first-order expansions which I believe might be due to my inability ...
14
votes
1answer
242 views

Why do big-O terms disappear in definite integrals since Mathematica 9?

In Mathematica 8, when I computed the following input: Integrate[Series[Cos[x], {x, 0, 2}], {x, 0, a}] Mathematica returned an expression that had a O[a^4] in ...
4
votes
0answers
48 views

FindSequenceFunction for sum of hypergeometric terms

Mathematica's built-in function FindSequenceFunction is quite good at recognizing hypergeometric terms, i.e. terms $c_k$ for which $c_{k+1}/c_k$ is a rational ...
3
votes
1answer
108 views

Problem with series expansion and integrate

I have the following very simple code. A power series in a-parameter, function f[x], is integrated with respect to x, and the ...
2
votes
1answer
57 views

Truncating power series

I have the following typed up so as to truncate higher powers of $\tau$, ...
2
votes
3answers
190 views

Extracting coefficients from a power series

I have a function defined explicitly as a power series: $$\sum_{n=0}^\infty{T_n}\frac{x^n}{n!}=\frac{\frac{x^3}{3!}}{e^x-1-x-\frac{x^2}{2!}}$$ and I would like to extract the coefficients $T_k$ as ...
1
vote
1answer
75 views

Simplify a series expansion including product and multiplication

I have a following expression $$ f=\Pi_{i=1}^n \left[ 1 + \frac{1}{t} +\frac{ (m+i)}{t^2} +\frac{ (m+i)^2}{t^3} + \cdots \right] $$ here $m,n,t$ are positive integers. I want to obtain a series ...
1
vote
1answer
64 views

Series expansion for rational function with weird powers of variable?

Consider the following series expression: Series[1/(a+b x^(1/3)+c x^(4/3)),{x,0,1}] The result comes out appropriately: 1/a - b x^(1/3)/a^2 +b^2 x^(2/3)/a^3 ...
1
vote
0answers
95 views

Asymptotics: replacing all arguments in a trig function with dummy variable after manipulation

I'm doing manipulations on functions of the form $$\phi(x,z,t) = \sum_n a_n(\epsilon x,\epsilon t,z)e^{in( k_ox -\omega_o t + \epsilon \theta(\epsilon x,\epsilon t)))} e^{(k_o+\epsilon k(\epsilon ...
1
vote
1answer
86 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
1answer
62 views

Series expanding an expression to an arbitrary power

How can I get Mathematica to directly produce the series expansion of an expression such as $(1+1/x)^n$, where $n$ is an arbitrary positive integer? Note that it's important in my expression to keep ...
2
votes
2answers
152 views

How to get the series coefficient of a function defined by a differential equation [duplicate]

I need a Taylor series approximation for $x(t)$, which is defined by the following differential equation. $\frac{dx}{dt}=-k_1 x+(1-x)k_2 e^{-k_3 t}$, $x(0)=0$ ...
3
votes
0answers
131 views

Generating function for Newton series?

The function GeneratingFunction gives generating function for Taylor series. Is there a similar function for Newton's series? $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)$$
6
votes
2answers
346 views

Finding the coefficient of a certain power in a generating function

I want to compute $(t^1+\dots +t^5)(t^2+\dots+ t^6)(t^3+\dots +t^9)$ and find the coefficient of $t^{15}$ for example. $t$ is an indeterminate Now in Maple this is simply as ...
0
votes
1answer
54 views

Speeding up ReplaceRepeated while truncating to desired order

I need to program in an algorithm that recursively makes algebraic replacements which leads to an utterly complicated algebraic function of $x$, but whose final result is only needed at fixed order in ...
2
votes
0answers
51 views

Implementation of Series function

I'm trying to evaluate the partial derivatives of some function F[x,y] at some point, i.e. ...
0
votes
1answer
123 views

Elegant way to do series expansion around x+$\epsilon$

Suppose I have a function $p(x+\epsilon,y-\epsilon,t)$, and I want to expand it around $(x,y)$ like $$ p(x+\epsilon,y-\epsilon,t)=p(x,y,t)+\dfrac{\partial p}{\partial x}\epsilon+\dfrac{\partial ...
0
votes
0answers
96 views

Best way to determine polynomial coefficients in series expansion

I would like to solve the equation $$h'(\boldsymbol{x}_1)\left[B_1\boldsymbol{x}_1+g_1(\boldsymbol{x_1},h(\boldsymbol{x}_1))\right]=B_2h(\boldsymbol{x}_1)+g_2(\boldsymbol{x}_1,h(\boldsymbol{x}_1))$$ ...
0
votes
1answer
43 views

Series not evaluating to numerical answer [closed]

I am working on an example from a book (An Engineer's Guide to Mathematica). The following code is in the book: ...
3
votes
1answer
146 views

Is there a function that, given a fraction, will return the general term of its infinite series expansion?

Is here some way to expand a fraction to an infinite sum in mathematica, i.e., a series? I want the general term of the series. For example, $\frac{2}{3(x-1)^3}$
0
votes
1answer
90 views

Series expansion for $\frac{1}{x+1}$ in terms of $\frac{1}{x-1}$

I would like to expand a function as $$\frac{1}{x+1} = \frac{1}{x-1+2} = \frac{1}{x-1} \frac{1}{ 1+\frac{2}{x-1}} = \frac{1}{x-1} \left[ 1- \frac{2}{x-1} + \left(\frac{2}{x-1}\right)^2 + \cdots ...
5
votes
1answer
106 views

Series expansion for $\frac{x}{1- \frac{1}{x}}$

I would like to expand $\frac{x}{1- \frac{1}{x}}$ as $$\frac{x}{1- \frac{1}{x}} = x \left( 1+ \frac{1}{x} + \frac{1}{x^2} +\frac{1}{x^3} + \cdots \right) = x + 1 + \frac{1}{x} + \frac{1}{x^2} + ...
1
vote
2answers
107 views

eccentric anomaly expansion equation

I try to use this simple algorithm (paper) to calculate the Eccentric Anomaly expansion: ...
0
votes
0answers
40 views

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
1
vote
0answers
65 views

Why does Mathematica not go to specified order in series?

For some reason Mathematica will not evaluate this asymptotic series to the requested order. Inputting: ...