268 CHAPTER 5 Series and Differential Equations
(a)∑∞
n=01
4 n^2 − 1(b)∑∞
n=03
9 n^2 − 3 n− 2- Consider the series
Σ =∑{ 1
n∣∣
∣∣
∣the integerndoesn’t contain the digit 0}
.Therefore, the series Σ contains reciprocals of integers, except that,
for example, 10 is thrown out, as is 20, as is 100, 101, etc. No 0s
are allowed! Determine whether this series converges. (Hint:Σ = 1 +1
2
+···+
1
9
+
1
11
+
1
12
+···+
1
19
+
1
21
+···+
1
99
+
1
111
+
1
112
+···+
1
999
+ ···
< 9 +
92
10
+
93
100
+···.)
- (Formal definition ofe) Consider the sequencean=
(
1 +1
n)n
, n=
1 , 2 ,....(a) Use the binomial theorem to show that an < an+1, n =
1 , 2 ,.... (Note that in the expansions of an and an+1, the
latter has one additional term. Moreover, the terms ofancan
be made to correspond to terms ofan+1with each of the terms
of the latter being larger.)(b) Show that, for each positiven,an<1+1+1
2!
+
1
3!
+···+
1
n!< 3.
(c) Conclude that limn→∞(
1 +1
n)n
exists. The limit is the familiar
natural exponential base,e, and is often taken as the formal
definition.
(d) Show that for any real number x, limn→∞Ç
1 +x
nån
=ex.(Hint:note that limn→∞Ç
1 +
x
nån
= limm→∞(
1 +1
m)mx
.)