Summary: We study the convergence of multiple harmonic series of the following form \[ \sum_{k_1,\ldots,k_n\geq 1} \frac {k^{l_1}_1 k^{l_2}_2\cdots k^{l_n}_n}{(k^{p_1}_1+ k^{p_2}_p+\ldots+k^{p_n}_n)^p} {\text{ and }} \sum_{k_1,\ldots,k_n\geq 1} \frac {(k^{l_1}_1 +k^{l_2}_2+\ldots+k^{l_n}_n)^m k^{b_1}_1 k^{b_2}_2\cdots k^{b_n}_n}{(k^{p_1}_1+k^{p_2}_p +\ldots+k^{p_n}_n)^p}, \] where \(m,l_i,b_i\) are nonnegative real numbers and \(p\) and \(p_i\) are positive real numbers. We determine exactly when the two harmonic series converge.