Tagged Questions

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

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0
votes
1answer
53 views

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

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
46 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
153 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
57 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
116 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
49 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
237 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
38 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
91 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
43 views

Truncating power series

I have the following typed up so as to truncate higher powers of $\tau$, ...
2
votes
3answers
149 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
47 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
47 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
83 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
52 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
46 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
144 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
121 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
285 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
49 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
49 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
101 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
77 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
42 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
141 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
85 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
96 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
94 views

eccentric anomaly expansion equation

I try to use this simple algorithm (paper) to calculate the Eccentric Anomaly expansion: ...
0
votes
0answers
31 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
61 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: ...
-1
votes
1answer
69 views

Plot CPU time vs iteration? [duplicate]

I would like to generate a plot of CPU time vs number of iterations. For example, if I were to calculate the solution of a system of differential equations in state-space form using the summation of ...
0
votes
1answer
110 views

How to calculate the unknown quantity in an infinite series?

I'd like to calculate x value in this equation. Basically, I tried to 2 types of method which are FindRoot and NSolve. But, I have failed the calculation caused by these errors up to now. If ...
0
votes
1answer
88 views

Problem of how to manipulate Taylor/McLaurin series

I'm new to Mathematica so I don't know very well the program itself. I would like to manipulate as he does this person in youtube: http://www.youtube.com/watch?v=fCJHvQaGNiQ But I put all the ...
1
vote
1answer
68 views

Power series with decimal and arbitrary powers

Is there a way to make Mathematica do a series of the form: Series[ E^{\beta}$ , {x, 0, 1}] I have noticed that ...
0
votes
0answers
28 views

Speed up series expansion [duplicate]

I have a rather complicated function that I need to expand it in series, which is quite straightforward but takes a long time. I was wondering if there is a way to speed up such a calculation: ...
0
votes
3answers
99 views

How to use Assumptions in a Series Expansion

I want to series expansion the expression $\frac{1}{2} \left(e_1+e_2-\sqrt{e_1^2-2e_1e_2+e_2^2+4V_{12}^2} \right)$ up to second order in $V_{12}$ using the assumption $e_1>e_2$. So I tried ...
4
votes
4answers
204 views

Exponential of a Differential Operator

In Mathematica, is it possible to exponentiate a differential operator such that the operator will act on a function, $f(x,p)$? Specifically, I wondering if I can get Mathematica to do this: ...
0
votes
1answer
100 views

How to power-series expand determinants?

Say $g$ is a 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 dimension as $g$) and ...
0
votes
1answer
86 views

How do I plot a Taylor polynomial of various degrees in two variables? [closed]

I want to plot the Taylor polynomials of $f(x,y)= Sin(1 + x + y^2)/(4 + x^2 + y^2)$ of degrees 4 and 7 around the point $(0,0)$ over the rectangle $[-\pi,\pi]\times [-\pi,\pi]$ I am currently using ...
0
votes
3answers
91 views

Create polynomials from Series

This question actually doesn't have quite a lot to do with the Series function, but I don't know how to describe my problem. So here's the thing. I'm trying to ...
8
votes
1answer
215 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}]
1
vote
1answer
110 views

divergent part of a series

I am trying to find a function that gives me the divergent part of series in any case. The method "Series[]" does not work because Series[42,{n,0,-1}] gives as ...
1
vote
1answer
54 views

Series and that old ivar error

I am trying to compare plots of the function (1+x)^{1/3} and its power series. The code for the power series generates an error message referring to an invalid variable: 'General::ivar: -1.09993 is ...
2
votes
1answer
106 views

Remove singularity at zero from $-((i^{-n} (-1 + i^n)^2)/(n^2 π^2))$

The expression h[n_] = -((I^-n (-1 + I^n)^2)/(n^2 π^2)) is real for all integers n. Although indeterminate at ...
0
votes
1answer
59 views

Plot Series output? [closed]

I tried to search if similar questions existed but I have not found anything. Can someone help me with this code? ...
1
vote
1answer
79 views

Working with means and variances

My goal is to work out on the following equation: $$Mean(P)(1-Mean (P))\left (h + (1 - 2 h) (1 - \frac {Var (P)} {Mean (P) (1 - Mean (P))}) (1 - Mean (P)) + \frac {Var (P)} {Mean (P) (1 - Mean ...
2
votes
1answer
106 views

Taylor expansion of function of vectors—simplifying the output

I've got a real-valued function of several vectors $f(u,v,w)$ formed by taking scalar products of linear combinations of the vectors, I want to Taylor expand around small $v$ by writing $$f(u,\delta ...
1
vote
1answer
64 views

Compose two special power series expansions

I have two functions $A(x), B(x)$, given in a special power series form: $A(x)=1-x^{2}\left(\frac{a}{10}-\sum_{k=1}^{9}b_{k}\left(\frac{(x^{2}-1)}{r}\right)^{k}\right)$ ...
2
votes
2answers
131 views

Bessel derivatives

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

A power series expansion

Consider the function, $f(z) = z tanh(\pi z) log (z^2 + a^2)$ for some $a>0$. Now I am considering 3 different situations, $z = i(n+0.5) - i\epsilon + \delta - it$ for $n \in \mathbb{Z}$ and ...