Find Laplace Transform using unit step function given graph of a periodic impulse function. (5.3-33) Please correct my work. The textbook answer which is expressed exactly like this $1/s(1+e^{-s})$ does not match my own.

Find the Laplace Transform for one period of the perpetual periodic function shown in graph. 
First express function using the unit step:
$$f(t)= u(t)-u(t-1)$$
The Laplace Transform is easily found:
$$f(t)=  \frac{1}{s}-\frac{e^{-s}}{s}= \frac{1-e^{-s}}{s}$$
Where did I go wrong?
 A: Here is a very simple method to solve this problem. Note that $f(t)-f(t-2)$ is just a single period of $f(t)$ in the range $[0,2]$, i.e.
$$f(t)-f(t-2)=u(t)-u(t-1)\tag{1}$$
the Laplace transform of which you've already found:
$$\mathcal{L}\{u(t)-u(t-1)\}=\frac{1-e^{-s}}{s}\tag{2}$$
But you also have
$$\mathcal{L}\{f(t)-f(t-2)\}=F(s)-e^{-2s}F(s)=F(s)(1-e^{-2s})\tag{3}$$
where $F(s)$ is the desired Laplace transform of the periodic function $f(t)$. Equating (2) and (3) gives
$$F(s)=\frac{1-e^{-s}}{s(1-e^{-2s})}=\frac{1}{s(1+e^{-s})}$$
because
$$\frac{a-b}{a^2-b^2}=\frac{a-b}{(a+b)(a-b)}=\frac{1}{a+b}$$
A: Your $f(t)$ is correct for the imagine provided if you only take one period.
$$
f(t) = \mathcal{U}(t)-\mathcal{U}(t-1)
$$
Now take the Laplace transform of 
$$
\int_0^{\infty}f(t)e^{-st}dt = \int_0^{1}e^{-st}dt
$$

If you want to find the Laplace transform of the periodic unit step function, then first we will write $f(t)$ as a series.
$$
f(t) = \sum_{n=0}^{\infty}\bigl[\mathcal{U}(t-2n)-\mathcal{U}[t-(2n+1)]\bigr]
$$
Let's write out a few terms.
\begin{align}
f(t) &= \mathcal{U}(t)-\mathcal{U}(t-1)+\mathcal{U}(t-2)-\mathcal{U}(t-3)+\mathcal{U}(t-4)-\cdots\\
& = \mathcal{U}(t) + \sum_{n=1}^{\infty}(-1)^n\mathcal{U}(t-n)\\
\mathcal{L}\{f(t)\} &= \int_0^{\infty}e^{-st}dt + \int_0^{\infty}\sum_{n=1}^{\infty}(-1)^n\mathcal{U}(t-n)e^{-st}dt
\end{align}
We can interchange the $\sum(-1)^n$ with integral since $\mathcal{U}(t-n)e^{-st}\geq 0$ for $t \geq 0$.
\begin{align}
&= \frac{1}{s} + \sum_{n=1}^{\infty}(-1)^n\int_n^{\infty}e^{-st}dt\\
&= \frac{1}{s} + \sum_{n=1}^{\infty}(-1)^n\frac{e^{-sn}}{s}\\
&= \frac{1}{s}\Bigl[1 - e^{-s}\sum_{n=0}^{\infty}(-1)^ne^{-sn}\Bigr]\\
&= \frac{1}{s}\Bigl[1 - \frac{e^{-s}}{1+e^{-s}}\Bigr]
\end{align}
