# Tag Info

### 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 ...
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### A more convenient Fourier series

For the reasons mentioned above, I wrote the following "shell" for these functions: ...
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### 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 ...
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### 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 $x(1-x)$...
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### 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: ...
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### 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 $\text{Re}(x)>0$: ...
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### 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 ...
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### 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)/...
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### 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 ...
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### 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 ...
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### 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 ...
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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} ... • 19.1k 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 ... • 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 ... • 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: ... 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 ... • 61.8k 10 votes Accepted ### Plotting a Taylor Series of two-variable trigonometric function Use Normal and Evaluate,e.g.: ... • 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 ... • 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 ... • 59.3k 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]); ... • 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 ... • 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]: ... • 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. ... • 3,352 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 ... • 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 $y=\sqrt[3]{z}$, $y=(-1)^{2/... • 497 9 votes ### Analytical approximation of indefinite integral on a given interval to a given precision One can construct a Chebyshev series approximation to the integrand for an interval, such as -5 <= x <=5 mentioned in the comments, and integrate it to get a ... • 238k 9 votes Accepted ### How can I get the minimum error term when manipulating Taylor series? You could just apply Series to the expression of interest: Series[(u[x+h] + u[x-h] - 2 u[x])/h^2, {h, 0, 3}] //TeXForm$u''(x)+\frac{1}{12} h^2 u^{(4)}(x)+O\...
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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. ...