Summation of a Slowly Convergent Series Arising in Antenna Study

An equivalent series for the slowly convergent series \[ \sum \limits _{n = 1}^\infty {\left [ {\smallint _{ - \pi /2}^{\pi /2}{{\cos }^\alpha }\theta \cos (n \in \sin \theta )} \right ]} {}^2/n\] which arises in antenna theory is obtained. The new form is found to consist of two rapidly convergent series for small $\in$.