310
CHAPTER 36
36.52
36.53
36.54
36.55
36.56
36.57
36.58
36.59
36.60
36.61
Show that a sequence may converge in the mean (Problem 36.51) without converging in the ordinary sense.
See Problem 36.2.
If
a n = L and each a > 0 (so that L > 0), prove that
Let b n = \na n . Then
So
by
Problem 36.51. Hence,
(The proof assumed only that L > 0. The result also can be
proved when L = 0 by a slight variation in the argument.)
Show that a n = 2n/(3n + 1) is an increasing sequence.
Then,
The
last inequality is obvious.
Determine whether «„ = (5n — 2)/(4n + 1) is increasing, decreasing, or neither.
Then
The last inequality is obvious, and, therefore, the sequence is increasing.
Determine whether a n =3"/(l + 3") is increasing, decreasing, or neither.
Then,
Since the last inequality is true, the sequence is increasing
Determine whether the sequence a n = n\!2" is increasing, decreasing, or neither.
Then,
Thus, the sequence is increasing for M > 1.
Determine whether the sequence a n = nil" is increasing, decreasing, or neither.
Then,
Thus, the sequence is
decreasing.
Determine whether the sequence a n = (n
2 - \)/n is increasing, decreasing, or neither.
Then,
Hence, the sequence is increasing.
Determine whether the sequence a n = n"ln\ is increasing, decreasing, or neither.
Then
Since the last inequality holds,
the sequence is increasing.
Determine whether the sequence
is increasing, decreasing, or neither.
Then,
So
and the sequence is
decreasing.