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Lebl-Basic-Analysis-I
· Problem 6.2.2
Sequences of Functions
Mathematics · uniform-convergence-derivative-limit
Problem
Let
f
n
(
x
)
:
=
x
n
n
f_n(x) \mathrel{:=} \dfrac{x^n}{n}
f
n
(
x
)
:=
n
x
n
. Show that
{
f
n
}
n
=
1
∞
\{f_n\}_{n=1}^\infty
{
f
n
}
n
=
1
∞
converges uniformly to a differentiable function
f
f
f
on
[
0
,
1
]
[0,1]
[
0
,
1
]
(find
f
f
f
). However, show that
f
′
(
1
)
≠
lim
n
→
∞
f
n
′
(
1
)
f'(1) \neq \lim_{n\to\infty} f_n'(1)
f
′
(
1
)
=
lim
n
→
∞
f
n
′
(
1
)
.
Related theory
Series Convergence Tests And Absolute Convergence
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