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Analysis 1: Exercises 15
1.
Suppose that
∑
∞
1
a
n
and
∑
∞
1
b
n
both converge. Let
α, β
∈
R
. Prove that
∞
X
1
(
αa
n
+
β b
n
) =
α
∞
X
1
a
n
+
β
∞
X
1
b
n
.
This property of series is called
linearity
.
2.
(a) Show that if
x >
0 and if
N
3
n >
2
x
, then
±
±
±
±
e
x

²
1 +
x
1!
+
···
+
x
n
n
!
³±
±
±
±
<
2
x
n
+1
(
n
+ 1)!
.
(b) Use the formula in (a) to show that 2
7
12
< e <
2
3
4
.
3.
Show that if 0
6
x
6
a
and
n
∈
N
, then
1 +
x
1!
+
···
+
x
n
n
!
6
e
x
6
1 +
x
1!
+
···
+
x
n

1
(
n

1)!
+
e
a
x
n
n
!
.
4.
Show that
log(
1 +
x
1

x
) = 2
²
x
+
x
3
3
+
x
5
5
+
x
7
7
+
···
³
for

x

<
1. The series is called
Gregory’s series
.
5.
Compute the following limits
(a) lim
x
→
0
a
x

1
x
where
a >
0;
(b) lim
x
→
0
log(1 +
ax
)
x
where
a >
0;
(c) lim
x
→
0
1

cos
x
x
2
;
(d) lim
x
→∞
´
cos
a
x
µ
x
2
, where
a >
0.
6.
(a) Let
f
: [0
,
∞
)
→
R
be given by
f
(
x
) =
x
log
x
for
x >
0 and
f
(0) = 0. Is
f
continuous at 0?
(b) Let
f
: [0
,
∞
)
→
R
be given by
f
(
x
) =
x
2
log
x
for
x >
0 and
f
(0) = 0. Is
f
continuous at 0? Is it rightdiﬀerentiable at 0?
7.
Prove that for
x
≥
0 the inequality
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 Winter '10
 Spyros

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