Mixing
Proving a sequence of functions converges, is differentiable, etc.
$begingroup$
I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative.
For each natural number $n$, define $f_n : [−1, 1] → R$ by
$$f_n(x) = frac{x}{1+n^2x^2}$$
(a) Prove that for each natural number $n$, the function $f_n$ is differentiable, and find $f'_n$
(b) Prove that $(f_n)$ is uniformly convergent.
(c) Prove that $(f_n)$ is pointwise convergent.
(d) Prove that $$(lim_{xtoinfty} f_n(x))' neq lim_{xtoinfty} f_n'(x)$$
I notice that the function is continuous and that $f'_{n}$ exists over the interval, so that sets me up well to prove that $f_n$ is differentiable.
I'm iffy on proving the last three points. Where do I begin?
real-analysis uniform-convergence pointwise-convergence
asked Jan 15 at 4:37
beflyguybeflyguy
354
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add a comment |
$begingroup$
I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative.
For each natural number $n$, define $f_n : [−1, 1] → R$ by
$$f_n(x) = frac{x}{1+n^2x^2}$$
(a) Prove that for each natural number $n$, the function $f_n$ is differentiable, and find $f'_n$
(b) Prove that $(f_n)$ is uniformly convergent.
(c) Prove that $(f_n)$ is pointwise convergent.
(d) Prove that $$(lim_{xtoinfty} f_n(x))' neq lim_{xtoinfty} f_n'(x)$$
I notice that the function is continuous and that $f'_{n}$ exists over the interval, so that sets me up well to prove that $f_n$ is differentiable.
I'm iffy on proving the last three points. Where do I begin?
real-analysis uniform-convergence pointwise-convergence
asked Jan 15 at 4:37
beflyguybeflyguy
354
$endgroup$
add a comment |
$begingroup$
I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative.
For each natural number $n$, define $f_n : [−1, 1] → R$ by
$$f_n(x) = frac{x}{1+n^2x^2}$$
(a) Prove that for each natural number $n$, the function $f_n$ is differentiable, and find $f'_n$
(b) Prove that $(f_n)$ is uniformly convergent.
(c) Prove that $(f_n)$ is pointwise convergent.
(d) Prove that $$(lim_{xtoinfty} f_n(x))' neq lim_{xtoinfty} f_n'(x)$$
I notice that the function is continuous and that $f'_{n}$ exists over the interval, so that sets me up well to prove that $f_n$ is differentiable.
I'm iffy on proving the last three points. Where do I begin?
real-analysis uniform-convergence pointwise-convergence
asked Jan 15 at 4:37
beflyguybeflyguy
354
$endgroup$
I have this question asking to prove that a sequence of functions is differentiable, pointwise and uniformly convergent, and something regarding the equality of limits of the function's derivative.
For each natural number $n$, define $f_n : [−1, 1] → R$ by
$$f_n(x) = frac{x}{1+n^2x^2}$$
(a) Prove that for each natural number $n$, the function $f_n$ is differentiable, and find $f'_n$
(b) Prove that $(f_n)$ is uniformly convergent.
(c) Prove that $(f_n)$ is pointwise convergent.
(d) Prove that $$(lim_{xtoinfty} f_n(x))' neq lim_{xtoinfty} f_n'(x)$$
I notice that the function is continuous and that $f'_{n}$ exists over the interval, so that sets me up well to prove that $f_n$ is differentiable.
I'm iffy on proving the last three points. Where do I begin?
real-analysis uniform-convergence pointwise-convergence
real-analysis uniform-convergence pointwise-convergence
asked Jan 15 at 4:37
beflyguybeflyguy
354
asked Jan 15 at 4:37
beflyguybeflyguy
354
asked Jan 15 at 4:37
beflyguybeflyguy
354
asked Jan 15 at 4:37
beflyguybeflyguy
354
asked Jan 15 at 4:37
beflyguybeflyguy
354
354
add a comment |
add a comment |
1 Answer
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$(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$
$(c)$ follows from $(b)$.
$(d)$ is now your turn !
answered Jan 15 at 5:46
FredFred
46.8k1848
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1 Answer
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1 Answer
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$begingroup$
$(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$
$(c)$ follows from $(b)$.
$(d)$ is now your turn !
answered Jan 15 at 5:46
FredFred
46.8k1848
$endgroup$
add a comment |
$begingroup$
$(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$
$(c)$ follows from $(b)$.
$(d)$ is now your turn !
answered Jan 15 at 5:46
FredFred
46.8k1848
$endgroup$
add a comment |
$begingroup$
$(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$
$(c)$ follows from $(b)$.
$(d)$ is now your turn !
answered Jan 15 at 5:46
FredFred
46.8k1848
$endgroup$
$(b)$: prove that $|f_n(x)| le frac{1}{2n}$ for all $n in mathbb N$ and all $x in [0,1].$
$(c)$ follows from $(b)$.
$(d)$ is now your turn !
answered Jan 15 at 5:46
FredFred
46.8k1848
answered Jan 15 at 5:46
FredFred
46.8k1848
answered Jan 15 at 5:46
FredFred
46.8k1848
answered Jan 15 at 5:46
FredFred
46.8k1848
46.8k1848
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