. Find the Maclaurin series for f(x) =

2022-05-12

. Find the Maclaurin series for f(x) = (1 + x) −3 using the definition. Find the radius of convergence.

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asked 2022-04-01

How to solve a cyclic quintic in radicals?
Galois theory tells us that
z111z1=z10+z9+z8+z7+z6+z5+z4+z3+z2+z+1 can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:
Let the roots be ζ1,ζ2,,ζ10, following Gauss we can split the problem into solving quintics and quadratics by looking at subgroups of the roots. Since 2 is a generator of the group [2,4,8,5,10,9,7,3,6,1] we can partition into the five subgroups of conjugate pairs [2,9],[4,7],[8,3],[5,6],[10,1].
A0=x1+x2+x3+x4+x5A1=x1+ζx2+ζ2x3+ζ3x4+ζ4x5A2=x1+ζ2x2+ζ4x3+ζx4+ζ3x5A3=x1+ζ3x2+ζx3+ζ4x4+ζ2x5A4=x1+ζ4x2+ζ3x3+ζ2x4+ζx5
Once one has A0,,A4 one easily gets x1,,x5. It's easy to find A0. The point is that τ takes Aj to ζjAj and so takes Aj5 to Aj5. Thus Aj5 can be written down in terms of rationals (if that's your starting field) and powers of ζ. Alas, here is where the algebra becomes difficult. The coefficients of powers of ζ in A15 are complicated. They can be expressed in terms of a root of a "resolvent polynomial" which will have a rational root as the equation is cyclic. Once one has done this, you have A1 as a fifth root of a certain explicit complex number. Then one can express the other Aj in terms of A1. The details are not very pleasant, but Dummit skilfully navigates through the complexities, and produces formulas which are not as complicated as they might be. Alas, I don't have the time nor the energy to provide more details.

asked 2022-04-10

What is the easier way to find the circle given three points?
Given three points (x1,y1),(x2,y2), and (x3,y3), if
y2y1x2x1y3y2x3x2y1y3x1x3,
then there will be a circle passing through them. The general form of the circle is
x2+y2+ dx +ey+f=0.
By substituting x=xi and y=yi, there will be a system of equation in three variables, that is:
(x1y11x2y21x3y31)(def)=((x12+y12)(x22+y22)(x32+y32)).
As there are a lot of things going around, the solution is prone to errors. Maybe this solution also has an error.
Is there a better way to solve for the equation of the circle?

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An MP3 manufacturer claimsthat 65% of teenagers 13 to 16 years old have their own MP3 players.
A researcher wishes to test the claim and selects a random sample of 80 teenagers. She finds that
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asked 2022-04-29

Find the Maclaurin series for the function f (x)  (x2  2  x)5 and its radius of convergence. 

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