Gauss's Test

If $u_n > 0$ and given $B(n)$ a bounded function of $n$ as $n\to \infty$, express the ratio of successive terms as

$$\left\vert{u_n\over u_{n+1}}\right\vert = 1 + {h\over n} + {B(n)\over n^r}$$

for $r>1$. The Series converges for $h>1$ and diverges for $h\leq 1$.

See also Convergence Tests

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 287-288, 1985.