Suppose that fa f (uniformly) and gn= g (uniformly) (pointwise) on A CR. on a set ACR. Show that fn9n → fg

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Do as directed.
(1)
Suppose that f =f (uniformly) and g, = g (uniformly) on a set A CR. Show that fn9n → fg
(pointwise) on A C R.
(ii) Provide an example to justify the statement "pointwise convergence does not preserve
boundedness." When can pointwise convergence preserve boundedness?
(iii) Show that an absolutely convergent series is convergent but converse is not true in general.
(iv) State and prove Abel's Test.
(v)
Show by Raabe's Test that the p-series converges if p > 1 and diverges otherwise.
Transcribed Image Text:Do as directed. (1) Suppose that f =f (uniformly) and g, = g (uniformly) on a set A CR. Show that fn9n → fg (pointwise) on A C R. (ii) Provide an example to justify the statement "pointwise convergence does not preserve boundedness." When can pointwise convergence preserve boundedness? (iii) Show that an absolutely convergent series is convergent but converse is not true in general. (iv) State and prove Abel's Test. (v) Show by Raabe's Test that the p-series converges if p > 1 and diverges otherwise.
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