Course Worksheet 9 - Differential and Integral Analysis PDF

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Teacher - Huy Nguyen...

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MATH 5105 Differential and Integral Analysis Exercise Sheet 9

Coursework Exercises 1. Consider the following sequence of functions fn . (a) fn (x) = nx , 1 (b) fn (x) = 1+x n ,

(c) fn (x) =

xn 1+xn .

For each of the functions above consider the following (i) Find f (x) = limn→∞ fn (x), (ii) Determine whether fn → f uniformly on [0, 1], (ii) Determine whether fn → f uniformly on R.

Problems 2. Prove the following: A sequence of functions {fn } on an interval I ⊆ R converges uniformly to a function f on I if and only if    lim [sup |fn (x) − f (x)| x ∈ I ] = 0 n→∞

Note that this means that by using calculus to determine that max of a function, we can determine if a sequence of functions is uniformly convergent. 3. Prove that if fn → f and gn → g uniformly on I ⊆ R then fn + gn → f + g uniformly. 4. Let fn (x) = x, gn (x) =

1 n

for all x ∈ R. Let f (x) = x, g(x) = 0.

(a) Show that fn → x, gn → 0 uniformly, (b) Does fn gn → fg uniformly?

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