9.2: Tests for Convergence

There are many ways to determine if a sequence converges—two are listed below. In all cases changing or removing a finite number of terms in a sequence does not affect its convergence or divergence:The Comparison Test makes sense intuitively, since something larger than a quantity going to infinity must also go to infinity. The Monotone Bounded Test can be understood by thinking of a bound on a sequence as a wall that the sequence can never pass, as in Figure [fig:mbtest]. The increasing sequence \(\seq{a_n}\) in the figure moves toward \(M\) but can never pass it. The sequence thus cannot diverge to \(\infty\), and it cannot fluctuate back and forth since it always increases. Thus it must converge somewhere before or at \(M\).4 Notice that the Monotone Bounded Test tells you only that the sequence converges, not what it converges to.Show that the sequence \(\seq{a_n}_{n=1}^{\infty}\) defined for \(n\ge 1\) by\[a_n ~=~ \frac{1 \,\cdot\, 3 \,\cdot\, 5 \,\cdots\, (2n-1)} {2 \,\cdot\, 4 \...