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Math Help - is this sequence convergent?

  1. #1
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    is this sequence convergent?



    Let Is the sequence convergent in ?
    First I tried to check if this sequence is Cauchy and I calculated integral of the sequence over . I got ). Then I'm stuck can anyone help?
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  2. #2
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    Quote Originally Posted by hermanni View Post


    Let Is the sequence convergent in ?
    First I tried to check if this sequence is Cauchy and I calculated integral of the sequence over . I got ). Then I'm stuck can anyone help?
    If the L_1-limit of the sequence f_n(t) = \frac1{\sqrt n}e^{-\frac1n(t-n)^2} exists, then it must be equal to the pointwise limit, which is the zero function. So all you need to do is to check whether \|f_n\|_1 = \displaystyle\int_0^\infty \!\! f_n(t)\,dt \to0 as n\to\infty. In fact, if you make the substitution u = \frac1{\sqrt n}(t-n), you see that \|f_n\|_1 = \displaystyle\int_{-\sqrt n}^\infty e^{-u^2}du \to\sqrt\pi, and so the sequence does not converge in the L_1-norm.
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    What about the same x_{n}  , is  { e^n x_{n} } is convergent in C[0,1] with sup norm? I think it converges to   e^{2t} .
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  4. #4
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    Quote Originally Posted by hermanni View Post
    What about the same x_{n}  , is  { e^n x_{n} } is convergent in C[0,1] with sup norm? I think it converges to   e^{2t} .
    I agree, the pointwise limit is   e^{2t} . To show that this is the limit in the sup norm, you need to show that e^n x_{n} \to  e^{2t} uniformly on [0,1]. So look at the difference e^n x_{n} -  e^{2t} and see if that can be made small for all t\in[0,1], provided that n is large enough.
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    One more thing : can you advise good functional analysis books ? (especially on banach spaces )
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