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If I have a function f(x) like this:

\begin{cases} 0 & 0 \leq x\leq1 \\ 1 & 1< x \leq 2 \end{cases}

I want to find sine series, for example so I use the formula below: enter image description here

My issue is, how to determine L? I assumed at the beginning that we look at the intervals, and see the amount in which they differ, like 1-0 = 1, or 2-1 = 1. But this is clearely wrong because in this prolem L =2. How do you determine L, in general?

Also, function graph:

enter image description here

1 Answers 1

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You are looking for $L$ such that $f(x+L)=f(x)$. The intervals of constancy are not important, it is the repeat.

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    Okay, thank you! but what is the technique to determine that? for example, am I expected to imagine the graph? – 2017-01-22
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    You are only given the function over a range of $2$ so you are using it all, or you are forcing it to repeat. Either way you know $L$ here. Other times it is not so clear. A graph can help – 2017-01-22
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    I'm using it all? I'm sorry, but I'm very confused by the explanation, but I attached the graph – 2017-01-22
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    As given your function is only given for x from 0 to 2. The graph you have drawn extends beyond that and has no period at all – 2017-01-22
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    this is part of the solution to reflect about the origin and extend the graph.. – 2017-01-22
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    anyway, I think I know what you mean now, thanks! – 2017-01-22