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24 votes

Series sum approximation

Writing: NSum[(-1)^n/Sqrt[Log[n]], {n, 2, Infinity}, Method -> "AlternatingSigns"] I get: 0.690243 which is what you want. In particular, directly from ...
πρόσεχε's user avatar
18 votes

A more convenient Fourier series

For the reasons mentioned above, I wrote the following "shell" for these functions: ...
xzczd's user avatar
  • 67k
16 votes

Chebyshev Approximation

Here's a way to leverage the Clenshaw-Curtis rule of NIntegrate and Anton Antonov's answer, Determining which rule NIntegrate selects automatically, to construct a ...
Michael E2's user avatar
  • 238k
16 votes

Find the function $f(x)$ by using its fourier expansion

Note that the expression returned by Sum is correct and equals $x(1-x)$ for $0 \leq x \leq 1$. I assume your question is how to simplify the expression into $x(1-x)$...
Greg Hurst's user avatar
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15 votes
Accepted

How to convert this term to a Hypergeometric function?

For your first question, if we gather the factors into a single variable z, there's a simple hypergeometric function: ...
Roman's user avatar
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14 votes

Help with Double Sum (lattice sum) over all integers m,n of 1/(a+m^2+n^2)

This kind of sum can be studied by integral transformation. Notice that $\int_0^1t^{x-1}dt=\frac{1}{x}$ if $\text{Re}(x)>0$: ...
Roman's user avatar
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13 votes

How to find the perturbation of $x^2 − 1 = \epsilon x$?

Decide up to which power you would like to expand: pow = 4; Let's do one of the equations you mentioned as an example (bring all terms to one side and save as a ...
Kagaratsch's user avatar
13 votes
Accepted

How to get the Taylor series of implicit functions

You can use AsymptoticSolve for this purpose: AsymptoticSolve[x+1/2y^2+1/2z+Sin[z]==0,{z,0},{{x,y},{0,0},4}] {{z -> -((2 x)/...
Carl Woll's user avatar
  • 131k
12 votes

Series vs Asymptotic in 12.1

Extended comment, I won't accept this as an answer. Here are some cases I've found where Series might be a better choice than ...
imas145's user avatar
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12 votes
Accepted

Build general form of an infinite sequence

...
Roman's user avatar
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11 votes
Accepted

How to represent $f(x) = (y-x)^k \log(y-x)$ as a summation of the form $f(x) = \sum\limits_{j=0}^\infty \cdots$?

Complete rewrite of answer The expression to be expanded as a series is the argument of ser = Series[(y - x)^k*Log[y - x], {x, 0, 5}] I attempted to obtain the ...
bbgodfrey's user avatar
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11 votes
Accepted

Why can’t mathematica find this residue?

You could use SeriesCoefficient instead: SeriesCoefficient[(z+1)^2 Exp[3/z^2], {z, 0, -1}] 6 Addendum Another possibility ...
Carl Woll's user avatar
  • 131k
11 votes
Accepted

Legendre expansion of the Dirac delta function

As I understood you start from the completeness relation $$\sum_{\ell=0}^\infty \frac{2\ell + 1}{2} P_\ell(x)P_\ell(y) = \delta(x-y)$$ and use that $$ P_n(0) = \begin{cases} \frac{(-1)^{m}}{4^m} ...
yarchik's user avatar
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11 votes
Accepted

Getting terms and only evaluate specific parts of a series

times[{i_}] := i times[{i__}] := Inactive[Times][i] Sum[times@Table[2 i - 1, {i, n}]/times@Table[2 i, {i, n}], {n, 5}] If you prefer ...
xzczd's user avatar
  • 67k
11 votes
Accepted

How to get more terms with the Series[] expansion of InverseErf[x] around x=1?

This is not strictly speaking an answer, but I thought I provide code that can be used to generate the kind of expansion that OP mentions. Perhaps it is useful for other people here. I use $y$ as an ...
user293787's user avatar
  • 11.9k
10 votes
Accepted

Speedier Algorithm or Set of Steps to Increase Speed of Coefficient Generation

It turns out the Olivier function is already built-in, but in a disguised form: ...
J. M.'s missing motivation's user avatar
10 votes
Accepted

Using Fourier Series to acquire Nonlinear ODE Periodic Solutions

Direct solution of the last equation in the question also is feasible, because the Fourier series converges very rapidly. as will be seen below. The equation for a three term expansion can be written ...
bbgodfrey's user avatar
  • 61.8k
10 votes
Accepted

Plotting a Taylor Series of two-variable trigonometric function

Use Normal and Evaluate,e.g.: ...
ubpdqn's user avatar
  • 62.1k
10 votes

Series expansion wrong

I think the behavior described is a bug, and I think it is related to the new enhanced support of Assumptions in Limit. In M11.1 ...
Carl Woll's user avatar
  • 131k
10 votes
Accepted

Trying to get a Laurent expansion of a symbolic function

A standard approach is to use a partial fraction decomposition, rearranged appropriately for the region of interest, that is, so that the corresponding infinite sum will converge. In this case the ...
Daniel Lichtblau's user avatar
10 votes
Accepted

How could this asymptotic expansion be obtained?

Exact expression for $\sigma_n$: b[n_] = BesselJ[1, BesselJZero[0, n]]*BesselJZero[0, n]*StruveH[0, BesselJZero[0, n]]; σ[n_] = π/2*(-1)^n*(b[n + 1] - b[n]); ...
Roman's user avatar
  • 48.4k
10 votes
Accepted

Error Message when nothing should be evaluated

This is due to the special behavior of SetDelayed (:=) with regards to the first argument (see e.g. this question): The ...
Lukas Lang's user avatar
  • 34.2k
10 votes

How to convert this term to a Hypergeometric function?

Match up power series and solve for parameters for Hypergeometric2F1[a, b, c, d x]: ...
Michael E2's user avatar
  • 238k
10 votes
Accepted

How to "prepare" expression for Taylor expansion

Both options give the expected result same result as Ulrich shows with their method, however, it can be seen that this is not to the second order that OP indicates they desire expanding to. ...
CA Trevillian's user avatar
10 votes
Accepted

Calculating relative error of Ramanujan formula for ellipse perimeter

We have to express a parameter $h=(a-b)^2/(a+b)^2$ in terms of the eccentricity of the ellipse $e = \sqrt{1-b^2/a^2}$. Similarly we need comparing the second Ramanujan approximation for the ...
Artes's user avatar
  • 57.5k
9 votes
Accepted

Finding an analytic solution of a cubic equation

It comes with which roots correspond to which branches of a cubic root in the exact expression. For instance, consider the simpler $$y^3 = z$$ Then my three solutions are $y=\sqrt[3]{z}$, $y=(-1)^{2/...
Alex Meiburg's user avatar
9 votes

Analytical approximation of indefinite integral on a given interval to a given precision

One can construct a Chebyshev series approximation to the integrand for an interval, such as -5 <= x <=5 mentioned in the comments, and integrate it to get a ...
Michael E2's user avatar
  • 238k
9 votes
Accepted

How can I get the minimum error term when manipulating Taylor series?

You could just apply Series to the expression of interest: Series[(u[x+h] + u[x-h] - 2 u[x])/h^2, {h, 0, 3}] //TeXForm $u''(x)+\frac{1}{12} h^2 u^{(4)}(x)+O\...
Carl Woll's user avatar
  • 131k
9 votes
Accepted

Removing higher order terms

You can use a variation of the idea I gave here: Normal @ Series[ ss /. {f:u1|u2 -> (s f[#1,#2,#3,#4]&)}, {s, 0, 3} ] /. s->1 u2[x, y, z, t]^...
Carl Woll's user avatar
  • 131k
9 votes

What series does Mathematica use for Hypergeometric1F1?

Recall that $(-n)_k=0$ for $k>n, n,k\in\mathbb N$. Thus, what you have is an appropriate truncation of the usual series for the Kummer function. ...
J. M.'s missing motivation's user avatar

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