Use the integral test to determine whether the infinite series is convergent. ∑n=1∞n−13 fill in the corresponding integrand and the value of the improper integral. enter inf for ∞, -inf for −∞, and dne if the limit does not exist. compare with ∫∞1 dx = by the integral test, the infinite series ∑n=1∞n−13a. convergesb. diverges

By the integral test, we can say that series [tex]\sum_{1}^{\infty }n^{-13}[/tex] is convergent.Given series is [tex]\sum_{1}^{\infty }n^{-13}[/tex]What is the integral test of convergence?As per the integral test of convergence:A series [tex]\sum_{1}^{\infty }n^{-13}[/tex]is convergent only if [tex]\int\limits^\infty_1 f(x)dx = \int\limits^\infty_1 {x^{-13} } \, dx[/tex] converges.[tex]\int\limits^\infty_1 f(x)dx = [\frac{x^{-12} }{-12} ]^\infty_1[/tex][tex]\int\limits^\infty_1 f(x)dx=\frac{1}{12}[/tex]Since [tex]\int\limits^\infty_1 f(x)dx[/tex] is finite i.e. 1/12 so integral converges.So, the series [tex]\sum_{1}^{\infty }n^{-13}[/tex] also converges.Therefore, By the integral test, we can say that series [tex]\sum_{1}^{\infty }n^{-13}[/tex] is convergent.To get more about the convergence of series visit: