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1. Prove that the series7n(a) converges uniformly on every bounded interval(b) does not converge absolutely for any value of r2. Suppose f is a real continuous function on R, n (t) (nt) and { n :n N) is equicontinuous on3. Suppose is an equicontinuous sequence of functions on a compact set K, and {fn) converges. Let K be a compact metric space, and S CC(K), where C (K) is the space of all complex continuous0,1. What conclusion can you draw about f?pointwise on K. Show that {fn converges uniformlyfunctions on K, equipped with the metric d ( .9) = max EK | (x)-g(z) . Suppose that s is closedand pointwise bounded. Show that S is compact5. Let {fn be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, andput岔Fn (x)/fn (t) dt for a ssb0Show that there exists a subsequence {Fn^1 which converges uniformly on [a, b

Question
1. Prove that the series
7n
(a) converges uniformly on every bounded interval
(b) does not converge absolutely for any value of r
2. Suppose f is a real continuous function on R, n (t) (nt) and { n :n N) is equicontinuous on
3. Suppose is an equicontinuous sequence of functions on a compact set K, and {fn) converges
. Let K be a compact metric space, and S CC(K), where C (K) is the space of all complex continuous
0,1. What conclusion can you draw about f?
pointwise on K. Show that {fn converges uniformly
functions on K, equipped with the metric d ( .9) = max EK | (x)-g(z) . Suppose that s is closed
and pointwise bounded. Show that S is compact
5. Let {fn be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, and
put
岔
Fn (x)/fn (t) dt for a s
sb
0
Show that there exists a subsequence {Fn^1 which converges uniformly on [a, b
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1. Prove that the series 7n (a) converges uniformly on every bounded interval (b) does not converge absolutely for any value of r 2. Suppose f is a real continuous function on R, n (t) (nt) and { n :n N) is equicontinuous on 3. Suppose is an equicontinuous sequence of functions on a compact set K, and {fn) converges . Let K be a compact metric space, and S CC(K), where C (K) is the space of all complex continuous 0,1. What conclusion can you draw about f? pointwise on K. Show that {fn converges uniformly functions on K, equipped with the metric d ( .9) = max EK | (x)-g(z) . Suppose that s is closed and pointwise bounded. Show that S is compact 5. Let {fn be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, and put 岔 Fn (x)/fn (t) dt for a s sb 0 Show that there exists a subsequence {Fn^1 which converges uniformly on [a, b

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Step 1

As per norms . three questions ,1a, 1b and 2 are answered. To analyze various aspects of convergence of the given series.

Step 2

1a)Proof that the given series converges for any real x (using Leibniz test on alternating series)

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Step 3

1a) Proof that the series is actually uniformly convergent provided x is in a bounded interval ...

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