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Showing posts with label MTH631 Assignment. Show all posts
Showing posts with label MTH631 Assignment. Show all posts
State Cauchy’s criterion and Weierstrass’s test for uniform convergence for series. Test the uniform convergence of the series `\sum u_n ; u_n = \frac{e^ -nx}{4n^2 - 1}` by using Weierstrass’s test.
Consider the sequence`{f_n}` of functions `f_n:R \to R` defined by `f_n (x) = \frac{nx}{\sqrt {1 + n^2x^2} }` Find the pointwise limit of this sequence as `n \to \infty.` Does the sequence converge uniformly on R? Justify it.
Using Weierstrass M-test determine the set on which the series `\sum 1/n^{3/2} (x/{1+x})^n` converges absolutely uniformly.
Consider the sequence of functions `(x) = 1/{1+x^n}` for `x \in [0,1]`. Check whether the sequence of functions `f_n (x)` converges uniformly on [0,1].
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