This question regards discovering an upper bound to the error in an approximation derived using the geometric mean.
I have two sequences of latent variables:
$$ r_1, r_2, ..., r_n$$ $$ s_1, s_2, ..., s_n$$
These latent variables combine to produce the observed variables, $R$ and $S$ (where $k$ is a scalar):
$$ R = (1+kr_1)(1+kr_2)..(1+kr_n)$$ $$ S = (1+ks_1)(1+ks_2)..(1+ks_n)$$
The true quantity, which is not observable, that I am interested in knowing is:
$$ T = (1+k(r_1+s_1))(1+k(r_2+s_2))..(1+k(r_n+s_n)) $$
In order to approximate $T$, I derive a geometric mean of the $r_i$ and $s_i$ by:
$$ (1+kr)^n = R \implies r = \frac{R^{1/n}-1}{k} $$ $$ (1+ks)^n = S \implies s = \frac{R^{1/n}-1}{k} $$
My approximation for $T$, $\bar{T}$, is:
$$ \bar{T} = (1+k(r+s))^n = (R^{1/n} + S^{1/n} - 1)^n$$
I would like to derive an upper bound, $U_b$ for the error, so that:
$$ | T - \bar{T} | \leq U_b $$
What can be said about $U_b$?