Exercise n 0.
Show that
1 − n 1
n
is bounded
0 ≤
1 − 1
n
n
< 1 Exercise n.1
Show that the sequence
x n =
1 − 1
n
n , is an increasing sequence. We have
x 2 > x 1 For n > 1 consider the ratio
x n+1
x n
=
1 − n+1 1
n+1
1 − n 1
n =
n n + 1
n+1 n n − 1
n
=
n n + 1
n+1 n n − 1
n+1 n − 1 n
=
n
(n + 1) n (n − 1)
n+1 n − 1 n
=
n 2 (n 2 − 1)
n+1
n − 1 n
= n 2 − 1 + 1 (n 2 − 1)
n+1
n − 1 n
= n 2 − 1 n 2 − 1 + 1
n 2 − 1
n+1 n n − 1
By Bernoulli inequality with
0 < h = 1
n 2 − 1 , ∀n > 1 Hence, substituing in the previous inequality
x n+1 x n
=
1 + 1
n − 1
n n − 1
> 1, ∀n > 1 Also, the sequence (x n ) defined by
x n = 1 − 1
n
n
, n = 1, 2, . . . .
is increasing by an application of inequality between geometrical and arithmetic mean. Indeed, applying the inequality
n+1
√
a 1 . . . a n+1 ≤ a 1 + · · · + a n+1 n + 1 with
a 1 = · · · = a n = 1 − 1
n and a n+1 = 1, we obtain
n+1