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August 16th, 2014, 06:09 PM  #1 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,540 Thanks: 2146 Math Focus: Mainly analysis and algebra  Convergent Sequence of Functions with NonZero Mean Square Error
I'm looking for an infinite sequence of functions, $f_n(x)$ that converge pointwise to zero on the closed interval $[0,1]$ but for which $$\lim_{n \to \infty} \int_0^1 f_n^2(x) \, \mathbb{d}x \ne 0$$ Any ideas? What about if the interval were open? 
August 17th, 2014, 08:25 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,540 Thanks: 2146 Math Focus: Mainly analysis and algebra 
\begin{align*}f_n(x) &= \begin{cases} n & \tfrac{1}{n} \le x \le \tfrac{1}{n1} \\ 0 &\text{otherwise} \end{cases} \\[12pt] \lim_{n \to \infty} \int_0^1 f_n^2(x) \,\mathbb{d}x &= \lim_{n \to \infty} n^2\left( \frac{1}{n1}  \frac{1}{n} \right) \\ &= \lim_{n \to \infty} n^2 \left( \frac{n  (n1)}{n(n1)} \right) \\ &= \lim_{n \to \infty} \frac{n^2}{n^2  n} \\ &= \lim_{n \to \infty} \frac{1}{1  \tfrac{1}{n}} \\ &= 1 \\ \end{align*} 
August 17th, 2014, 03:35 PM  #3 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 131 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus 
Why not $\displaystyle f_{n}(x)=\begin{cases}n &,x\in\left(0,\frac{1}{n}\right)\\0 &,\text{otherwise}\end{cases}$?

August 17th, 2014, 05:44 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,540 Thanks: 2146 Math Focus: Mainly analysis and algebra 
It's not straightforward to show that your function tends to zero at each point of $[0,1]$. I'm tempted to say that for every $n$ there is a region $(0,a)$ for which $f_n(x) \ne 0$. For any $c \in (0, 1]$ my sequence has $f_n(c) \ne 0$ for only one value of $n = N_c$. So for $n \gt N_c$ we have that $f_n(c) = 0$ and thus $f_n(x) \to 0$ for all points in $[0,1]$. I suppose for your sequence you'd say that for each $c \in (0,1)$ there exists $N_c$ such that $f_n(c) = 0$ for all $n \gt N_c$, but there's still that region to the left that is nonzero. It's a conceptually more difficult solution. 
August 18th, 2014, 02:17 AM  #5 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 131 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus 
$f_n(0)=0$ (we are in the "otherwise" category)


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convergent, error, functions, nonzero, sequence, square 
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