divergent.EXAMPLE 29
Determine whether converges absolutely, converges conditionally, or diverges.SOLUTION: Note that, since the series passes the nth Term Test.But is the general term of a convergent p-series (p = 2), so by the Comparison Test the
nonnegative series converges, and therefore the alternating series converges absolutely.EXAMPLE 30
Determine whether converges absolutely, converges conditionally, or diverges.
SOLUTION: is a p-series with so the nonnegative series diverges.
We see that
so the alternating series converges; hence is conditionally convergent.APPROXIMATING THE LIMIT OF AN ALTERNATING SERIES
Evaluating the sum of the first n terms of an alternating series, given by yields an
approximation of the limit, L. The error (the difference between the approximation and the true limit)
is called the remainder after n terms and is denoted by Rn. When an alternating series is first shown
to pass the Alternating Series Test, it’s easy to place an upper bound on this remainder. Because the
terms alternate in sign and become progressively smaller in magnitude, an alternating series
converges on its limit by oscillation, as shown in Figure N10–1.
FIGURE N10–1
Error bound
Because carrying out the approximation one more term would once more carry us beyond L, we