Let {fn}=1 be a sequence of continous functions on an interval [a, b]. If {fn}n=1 converges uniformly to a function f on [a, b], then f is continous on [a, b]. (a) (i) Discuss the convergence of the sequence of functions fn, when fn(x) = 1+ xn' where r E [0, 1].
Let {fn}=1 be a sequence of continous functions on an interval [a, b]. If {fn}n=1 converges uniformly to a function f on [a, b], then f is continous on [a, b]. (a) (i) Discuss the convergence of the sequence of functions fn, when fn(x) = 1+ xn' where r E [0, 1].
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Can I have a detailed, step-by-step explanation for part (a)-1 of the following question?
Kindly include the relevant reasoning/assumptions made during simplifications & calculations along with them.
Thank you very much!
![Let {fn}1 be a sequence of continous functions on an interval [a, b]. If {fn}1 converges
uniformly to a function f on [a, b], then f is continous on [a, b].
(a) (i) Discuss the convergence of the sequence of functions fn, when fn(x)
1+ xn
where r E [0, 1].
(ii) Show that the sequence {fn}, where fn(x) =
1
is not uniformly convergent
1+ xn
on r E (0, 1].
(b) Let fn(x) =
is not uniform.
14 „2n X E [0, 1]. Use the above theorem to show that the convergence
(c) Is the converse of the above theorem necessary true? Justify your answer using the
sequence fn(x) = nxe-n, x E [0, 1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb27e8b87-bff8-4dfb-b0ff-38ce7d501282%2Fa3e3d6b0-b83c-429c-a2c1-3a0ef0f9bb07%2Fpfbnq59_processed.png&w=3840&q=75)
Transcribed Image Text:Let {fn}1 be a sequence of continous functions on an interval [a, b]. If {fn}1 converges
uniformly to a function f on [a, b], then f is continous on [a, b].
(a) (i) Discuss the convergence of the sequence of functions fn, when fn(x)
1+ xn
where r E [0, 1].
(ii) Show that the sequence {fn}, where fn(x) =
1
is not uniformly convergent
1+ xn
on r E (0, 1].
(b) Let fn(x) =
is not uniform.
14 „2n X E [0, 1]. Use the above theorem to show that the convergence
(c) Is the converse of the above theorem necessary true? Justify your answer using the
sequence fn(x) = nxe-n, x E [0, 1].
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