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I need to find Fourier transform of

$$\sum_{n=-\infty }^\infty e^{-|t-2n|}$$

I found the solution online. But I think the choice of limits in equation (1) of the solution is wrong.It should be probably from -infinity to 2n for the first integral and from 2n to infinity for the second integral.But I am confused about what the limits would be as the both integration and summation is involved. and I am not sure whose limits to change(Probably limits of integration must be changed only and that of summation must be kept as it is)

Given Solution:

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Please tell me what the limits will be and the explanation. Also if possible please solve the problem.

Original answer:

$$ \frac{(1-e^{-2(1+jw)})}{(1-e^{-2})(1+jw)}-\frac{(e^{-2})(1-e^{-2(1+jw)})}{(1-e^{-2})(1-jw)}$$

OR

Answer:

enter image description here

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    Through away your reference: it is completely erroneous, because if you take the inverse Fourier of the result, you will get something of the form $Ae^{-B|t|}$ without any summation ! – 2017-02-13
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    Agreed Sir.... But is the 'original answer' to the problem that I have provided below the online solution, Is that correct? – 2017-02-14
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    I hadn't noticed it. It looks the desirable form. Sorry no time to check it. It would be advantageous to group the 2 fractions with a unique denominator $1+\omega^2$ in order to have sine/cosine functions. – 2017-02-14
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    I have done this exploiting the time shifting property of F.T:F.T of $$ e^{-|t|}= X_1(jw)= \frac{2}{1+w^2}$$ $$ x(t)=...+e^{-|t+4|}+e^{-|t+2|}+e^{-|t|}+e^{-|t-2|}+e^{-|t-4|}+...$$ <----F.T----> $$...+\frac{e^{jw4}}{1+w^2}+\frac{e^{jw2}}{1+w^2}+\frac{e^0}{1+w^2}+\frac{e^{jw4}}{1+w^2}+\frac{e^{jw4}}{1+w^2}+... $$ $$=\sum_{n=-\infty}^\infty \frac{e^{-jw2n}}{1+w^2}$$ – 2017-02-15
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    But the given function is a periodic function. And we know that Fourier transfom of periodic signal consists of impulses.This does not look like a function consisting of impulses. Where might have I gone wrong? – 2017-02-15

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