- #1
Felafel
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Homework Statement
study the pointwise and the uniform convergence of
##f_{n1}(x)=ln(1+x^{1/n}+n^{-1/x}## with ##x>0## , ##n \in |N^+}## and ##f_{n2}(x)=\frac{x}{n}e^{-n(x+n)^2}## with ##x \in \mathbb{R} ## , ##n \in }|N^+}##
The Attempt at a Solution
1) first series: ##f_{1n}##
studying the limit for n to infinity i found out it is ln2, so it converges pointwise to this value, but being the function increasing it doesn't have a maximum, and thus Weierstrass' criterion for uniform convergence doesn't yield.
however, for any compact in (0, infinity), say [a,b] with b>a it has a maximum in b. Thus, the function is also uniformly convergent for any compact in (0, infinity) but not in all ##\mathbb{R}^+##.
2) second series: ##f_{2n}##
studying the limit for n to infinity i found out it is 0, so it converges pointwise to this value.
am I missing some parts of the study/did I make any mistakes?
thank you in advance!
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