Analysis of hyperscaling in the Ising model by the high-temperature series method

High-temperature series expansions are derived for ($\frac{{\partial }^2\chi }{\partial H^2}$) for arbitrary dimension. These series and others already available are analyzed to determine if hyperscaling holds. Hyperscaling is found to hold for $d=2\mathrm{}$ but fails in three and four dimensions as $2\Delta -d\nu -\gamma =-0.028\mathrm{}\pm 0.003 \mathrm{}(d=3\mathrm{})\mathrm{}$, $-0.302\mathrm{}\pm 0.038 \mathrm{}(d=4\mathrm{})\mathrm{}$. The triviality of hyper-strong-coupling, Euclidean, boson, ${:{\phi }^4:}_{3,4}$ field theory follows. The expected location of the zero of the $\beta$ function in the Callan-Symanzik equation is computed in two and three dimensions.