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uniform convergence of $f_n(x)=\dfrac{n^3x^2}{(1+nx)^2}$
This is my first post, so let me begin by explaining what I am trying to do.
I have no idea how to find the uniform limit of $$f_n(x)=\dfrac{n^3x^2}{(1+nx)^2}$$
I’ve spent some time playing around with different constant $\alpha$ and comparing it to $x^2$ but it hasn’t given me much insight.
A:
As you said, you don’t need to deal with this, but it has two series expansions:
$$f_n(x)=\dfrac{n^3x^2}{(1+nx)^2}=\dfrac{n^3x^2}{2(nx)^2+n^2x^2}=\dfrac{n^3x^2}{2(nx)^2+n^2x^2}=\dfrac1{2}\left(\dfrac1{nx}-\dfrac1{n^2x^2}\right)$$
For $0\le x \le 1$ the series is absolutely and uniformly convergent by the Weirstrass M-test.
This could be extended to any $[0,\infty]$, but you get no uniform convergence.
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DISCLAIMER: All files are copyrighted, please don’t distribute without permission.Identification and characterization of a Xenopus laevis RTP1L-like gene with putative phospholipid binding sites.
We have identified a new gene (PhoLr) in the X. laevis genome database that shows sequence homology to RTP1L, a gene previously implicated in phospholipid and macrophage migration. We show that PhoLr is expressed as two transcripts that originate from two promoters and are differentially regulated during embryogenesis. An enhanced green fluorescent protein fusion with the PhoLr isoform that is initially expressed in the ectoderm at the gastrula stage encodes a cytosolic protein that is relocalized in granular structures and vesicles associated with the cell cortex during gastrulation. These cellular localizations are similar to those of RTP1L-like proteins from mouse and the Drosophila homolog. As in the case of the Xenopus RTP1L gene, phoLr mRNA is also expressed in the notochord at later stages, and its expression is down-regulated after the activation of the bmp signaling pathway, which is necessary for notochord specification. PhoLr mRNA also accumulates in the tadpole neurula, where it is localized in somatic and neuronal cells. Furthermore, antibodies against Xenopus PhoLr recognize a 54 kDa protein in detergent extracts of notochordal tissue
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