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5

Clear[a] a[0] = 0; a[k_] := a[k] = e*Sin[u]*(1 - a[k - 1]^2/2 + a[k - 1]^4/24) + e*Cos[u]*(a[k - 1] - a[k - 1]^3/6) // TrigReduce // Expand; TraditionalForm /@ a[2] TraditionalForm /@ (a[2] // Collect[#, Table[Sin[n*u], {n, 5}]] &)


3

There seem to be several issues here. The Mathematica code seems strange, arguably miswritten, though it appears you transcribed it correctly. Simplify[a[k]]; doesn't appear to do anything; it's output is never assigned Placing TeXForm[a[k]] >>"tex.01"; within the loop writes that file, then overwrites it in the next iteration. Perhaps ...


2

For any square matrix M which is the sum of two similar matrices M = A + B the determinant can be written as a sum of determinants as follows (example for two dimensions): det(M) = det( ( A11 + B11, A12 + B12), (A21 + B21, A22 + B22) ) = det( ( A11 + 0, A12 + 0), (A21 + B21, A22 + B22) ) + det( ( 0 + B11, 0 + B12), (A21 + B21, A22 + B22) ) and, ...


1

How about Table[First@Timing[N[Sum[(MatrixPower[A, j] t^j)/j!, {j, 0, i}]]], {i, 0, 200}] ListLinePlot[%] If you want the iteration number side by side, just change the body of the Table to {i, First@...}.


5

There are a few problems with your code. Allow me to highlight them and guide you to a solution. First of all, the proper way to define a function in Mathematica is using :=. So your code should read F[x_] := NSum[Exp[-x BesselJZero[0, a]^2]/BesselJZero[0, a]^2, {a, 1, Infinity}] Furthermore, you should note that the zeroes of the Bessel function are ...


1

Manipulate[ Plot[{Sin@x, Normal@Series[Sin@u, {u, x0, n}] /. u -> x}, {x, -2 Pi, 2 Pi}, PlotRange -> {Automatic, {-2, 2}}, Epilog -> {PointSize[Medium], Point@{x0, Sin@x0}}], {n, 0, 10, 1}, {x0, -Pi, Pi}]


0

This is not an answer but a general note. for the case of this function: E^x^(1/2) you have to note that the derivative of E^x^(1/2) at x=0 is ComplexInfinity. to show you that, the general series of a function is as follows: Clear[f] s = Series[f[x], {x, 0, 1}] // Normal if you set f[x] to your function it will result in ComplexInfinity. f[x_] ...



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