(E) Now the first omitted term, , is positive for x < 0. Hence P 9 (x) is
less than tan−^1 x.
(E) Each series given is essentially a p-series. Only in (E) is p > 1.
(C) Use the Integral Test in Chapter 10.
(C) The limit of the ratio for the series is 1, so this test fails; note
for (E) that
.
(B) does not equal 0.
(C) Since , the series converges if |x| < 1. We must test the
endpoints: when x = 1, we get the divergent harmonic series; x = −1
yields the convergent alternating harmonic series.
(A) for all x ≠ −1; since the given series converges to 0 if x =
−1, it therefore converges for all x.
(B) The differentiated series is ; so
.
