Sequences of Functions

We can construction a sequence of functions in more than one way. We can define a sequence of pointsand then define a sequence to bebut this is only actually a sequence of points since each f(x-n) is just a point and not a true function. A true function from which we may construct a sequence is actually a function of both n and x, where n takes all values in the natural numbers. For example we may define

For each sequence so defined we are especially interested in what happens as The sequence may converge to a well defined function, or may not converge. Convergence may depend on the set of values ofor on a particularIfis defined on the intervalthen ifas and iffor allWe say thatconverges to

The sequence converges to different points for different values ofIf we had definedon the intervalthen the sequence would not have converged because ifas

If the sequenceconverges to a functionit may converge uniformly or pointwise. If a function converges uniformly then

for all

The sequencedoes not converge uniformly since ifthenandhenceand

Henceis pointwise convergent but not uniformly convergent to

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