document.write( "Question 438196: Let $\"f%5Bn%5D%28x%29+=+nxe%5E%28-nx%5E2%29\"$ , for x in [0,1]. Show whether or not
\n" ); document.write( "i) $\"f%5Bn%5D%28x%29+\"$ converges for each x in [0,1].
\n" ); document.write( "ii) $\"f%5Bn%5D%28x%29\"$ converges uniformly on [0,1]. \n" ); document.write( "

Algebra.Com's Answer #303110 by robertb(5830) $\"\"$ $\"About$

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i) Fix any x-value in the interval (0,1].
\n" ); document.write( "Now $\"f%5Bn%5D%28x%29+=+nxe%5E%28-nx%5E2%29+=+%28nx%29%2Fe%5E%28nx%5E2%29\"$ \r
\n" ); document.write( "\n" ); document.write( "==>

\r
\n" ); document.write( "\n" ); document.write( "= $\"+x%2Alim%28n-%3Einfinity%2C+1%2F%28x%5E2e%5E%28nx%5E2%29%29%29\"$ by L'Hopital's rule.\r
\n" ); document.write( "\n" ); document.write( "= $\"%281%2Fx%29%2Alim%28n-%3Einfinity%2C+1%2Fe%5E%28nx%5E2%29%29+=+%281%2Fx%29%2A0+=+0\"$ \r
\n" ); document.write( "\n" ); document.write( "If x = 0, $\"f%5Bn%5D%280%29+=+%28n%2A0%29%2Fe%5E0+=+0\"$ for all n.\r
\n" ); document.write( "\n" ); document.write( "Hence for all x in [0,1], the sequence $\"f%5Bn%5D%28x%29\"$ exhibits pointwise convergence, and $\"lim%28n-%3Einfinity%2C+f%5Bn%5D%28x%29%29+=+0\"$ .\r
\n" ); document.write( "\n" ); document.write( "ii) Now $\"df%5Bn%5D%28x%29%2Fdx+=+%28n+-+2n%5E2x%5E2%29%2Fe%5E%28nx%5E2%29\"$ .
\n" ); document.write( "Setting this derivative to 0 and solving for x, we get $\"x%5E2+=+1%2F%282n%29\"$
\n" ); document.write( "<==> $\"x+=+1%2Fsqrt%282n%29\"$
\n" ); document.write( "Implementing the 1st derivative test using the test points $\"1%2Fsqrt%282%28n%2B1%29%29\"$ and $\"1%2Fsqrt%282%28n-1%29%29\"$ , we find that there is (absolute) maximum at $\"x+=+1%2Fsqrt%282n%29\"$ .
\n" ); document.write( " (Remember that $\"1%2Fsqrt%282%28n%2B1%29%29+%3C+1%2Fsqrt%282n%29+%3C+1%2Fsqrt%282%28n-1%29%29\"$ .)
\n" ); document.write( "Then $\"f%5Bn%5D%281%2Fsqrt%282n%29%29+=+sqrt%28n%2F%282e%29%29\"$ . Hence as $\"n+-%3E+infinity\"$ , the maximum of the function goes to infinity, and so the sequence of functions $\"f%5Bn%5D%28x%29\"$ does not exhibit uniform convergence.
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