Robert Israel
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 Jan 27 comment An intuitive explanation for the negatives of divergent summations? Of course I wasn't evaluating the same summation. I'm simply saying that "it comes out negative" is not any kind of general rule. You have so far given two examples where a summability method gives a negative result. It's like looking at Condaleeza Rice and Hilary Clinton and asking "why is every Secretary of State female?" Jan 27 answered An intuitive explanation for the negatives of divergent summations? Jan 27 comment integrals with no analytic answer - intuition and proof Certainly not that. Constant functions are elementary, so every real number is a value of an elementary function at a rational $x$. Jan 27 comment integrals with no analytic answer - intuition and proof Of course there are various methods that sometimes produce such a closed form. When none of the known methods apply, and numerical evaluation produces a number that isn't easily recognizable, we might strongly suspect that there is no closed form, but to prove such a statement (for any reasonable definition of "closed form") is essentially hopeless. Jan 27 comment integrals with no analytic answer - intuition and proof AFAIK very little is known in general about closed forms for definite integrals of elementary functions. Jan 26 comment Numerical integration of a system of stiff ODEs starting at a singular point If your trajectory is going to be in the singular plane, you can treat that as a constraint: substitute $x_1 = x_2$ and have two nonsingular differential equations in the two unknowns $x_2, x_3$. Jan 26 answered Eccentricity of a general ellipse Jan 26 comment Proof for the length of plane curves @Jasser That's just the convergence of Riemann sums to the integral. Note that $\sqrt{f'(t)^2 + g'(t)^2}$ is continuous since $f'$ and $g'$ are continuous. Jan 26 comment Finding a Matrix of Rank 10 using Taylor Expansion The Lagrange form for the remainder in Taylor's theorem. Note that $f^{(k+1)}(\xi) = e^\xi < e$ if $-1 < \xi < 1$. Jan 26 comment Finding a Matrix of Rank 10 using Taylor Expansion You are finding a matrix. The matrix has the entries $B_{ij}$. Jan 26 comment Finding a Matrix of Rank 10 using Taylor Expansion 10 terms of the Taylor series for $\exp(t_i t_j)$. Each $k$ gets one of them. Jan 26 comment Finding a Matrix of Rank 10 using Taylor Expansion Which part don't you understand? Jan 26 revised Finding a Matrix of Rank 10 using Taylor Expansion added 1 character in body Jan 26 answered At least $P(m, n - 1) = {{m!}\over{(m - n+1)!}}$ surjective functions from $[m]$ to $[n]$? Jan 25 answered Finding a Matrix of Rank 10 using Taylor Expansion Jan 25 answered How to prove that $A^TA$ is singular for $2\times 3$ matrix $A$ Jan 25 comment An identity involving binomial coefficients Please don't destroy your question. Others may find it and its answer useful. Jan 25 revised Defining Unitary Matrices added 85 characters in body Jan 25 answered Defining Unitary Matrices Jan 25 answered Proof for the length of plane curves