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(Solved): (a) Let \( \left\{f_{n}\right\} \) be a sequence of functions defined on a nonempty subset \( D \) ...



(a) Let \( \left\{f_{n}\right\} \) be a sequence of functions defined on a nonempty subset \( D \) of \( \mathbb{R} \) and le

(a) Let \( \left\{f_{n}\right\} \) be a sequence of functions defined on a nonempty subset \( D \) of \( \mathbb{R} \) and let \( x_{0} \in D \). Also, let \( f \) be a function on \( D \). Define what is meant by saying that \( \left\{f_{n}\right\} \) converges (i) point wise (ii) uniformly to \( f \) on \( D \). (b) Let \( \left\{f_{n}\right\} \) be the sequence of functions on \( (0,1) \) defined by \( f_{n}(x)=\frac{n^{2} x}{1+n^{3} x^{2}} \). Show that \( \left\{f_{n}\right\} \) is poinwise convergent but not uniformly convergent. (c) Show that the series \( \sum_{n=0}^{\infty}\left(1+x^{4}\right)\left\{1-\frac{1}{\left(1+x^{4}\right)^{n}}\right\} \) does not converge uniformly over \( (0, k] \), where \( k>0 \).


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Given: The sequence {fn}is the sequence of functions defined on a non-empty subset Dof Rand let x0?
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