# Sequence equivalent research

I'm studying this series:

I know that there is this relation:

($\ell$ is the common limit of the two series)

Do you know a Mathematica function that could prove this relation?

• You should look at RSolve , although on first pass it doesn't seem to be able to solve this. Feb 20 '18 at 19:43
• I think you need a0,b0 to have the same sign BTW. Feb 20 '18 at 19:49
• Yes a0 and b0 are > 0. But, did you manage to have the result? Feb 20 '18 at 20:53
• @george2079 I had a new picture with a new relation that might help. Feb 20 '18 at 20:56
• I confirmed your result numerically. That's not a proof though. re: edit you should see what RSolve does with that. Feb 20 '18 at 22:39

This is not a proof but rather an empirical test

Clear[a, b]

#[{a, b}] & /@ {Mean, HarmonicMean, GeometricMean}

(* {(a + b)/2, 2/(1/a + 1/b), Sqrt[a b]} *)


Since the terms converge, then you can used FixedPoint to find the limit

gm[a_?Positive, b_?Positive] :=
FixedPoint[{Mean[#], HarmonicMean[#]} &, {a, b}]


Testing whether the limits are the same and equal to the GeometricMean for 10,000 pairs of random reals

And @@ Table[
{a, b} = RandomReal[{10^-9, 100}, 2, WorkingPrecision -> 15];
g = gm[a, b];
g[[1]] == g[[2]] == GeometricMean[{a, b}],
{10000}]

(* True *)


You can use FixedPointList to look at the convergence step-by-step

FixedPointList[{Mean[#], HarmonicMean[#]} &, {5., 79.}]

(* {{5., 79.}, {42., 9.40476}, {25.7024, 15.3682}, {20.5353, 19.2352},
{19.8852, 19.864}, {19.8746, 19.8746}, {19.8746, 19.8746},
{19.8746, 19.8746}, {19.8746, 19.8746}} *)

GeometricMean[{5., 79.}]

(* 19.8746 *)

• I’m sorry but I don’t understand how it demonstrate the equality that I have to show ? Feb 21 '18 at 23:27
• This shows that the common limit of the two series is the GeometricMean, i.e., l = Sqrt[a0*b0] Feb 22 '18 at 1:07