TVM solve for interest rate, revisited

[Albert Chan]

(06-19-2022 06:57 PM)Albert Chan Wrote:  Let R = 1+x > 0, to prove:

g = n*(R-1)/(R^n-1) ≥ 1 - (n-1)/2*(R-1)

→ Or, n ≥ (1 - (n-1)/2*(R-1)) * (R^n-1)/(R-1)      // note: multiplied factor > 0

(RHS, R)' = (1 - n*(n+1)/2*R^(n-1) + (n²-1)*R^n - n*(n-1)/2*R^(n+1)) / (R-1)^2

If n > 1, numerator have 3 sign changes, 1 or 3 positive roots (Descartes rules of signs)
Taking taylor series, we showed that it had triple equal roots, at x=0 ...

I was being stupid.

Multiplied factor is a polynomial = (R^(n-1) + R^(n-2) + ... + R + 1)
In other words, RHS is just a polynomial. All its derivatives also polynomial.

(RHS,R)' roots = (1 or 3) - 2 = (-1 or 1) = 1 positive root --> 1 extremum (*)

No CAS needed for the proof

(*) we can say extremum must be where curve and tangent line touched.
In other words, numerator M have triple equal roots, at R = 1

Or, we do the math, and confirm M's 3 roots:

M = 1 - n*(n+1)/2*R^(n-1) + (n^2-1)*R^n - n*(n-1)/2*R^(n+1)

M(R=1) = 1 - n*(n+1)/2 + (n^2-1) - (n^2-n)/2 = 0      // M first root at R=1
M' = -(n+1)*n*(n-1)/2 * R^(n-2) * (R-1)^2                  // other 2 roots, also R=1