So basically, in my textbook I am given this way of majoring and minoring a series"
Let $f: [0,+\infty)$, strictly decreasing on the interval.
Let $k \in \mathbb{N}$, then we have:
$$\int^{k+1}_{k}f(t)dt \: \leq \: f(k) \:\leq \:\int^{k}_{k-1}f(t)dt$$
So far so good.
Now, I am given this:
$$f(0) + \int^{n+1}_{1}f(t)dt \: \leq \sum^{n}_{k=0}f(k) \: \leq \: \:f(0)+\int^{n}_{0}f(t)dt $$
What I fail to understand is the reason the $f(0)$ are added on both sides
If we subtract $f(0)$ from all sides, then
$$\int_1^{n+1}f(t)\ dt\le\sum_{k=1}^nf(k)\le\int_0^nf(t)\ dt$$
Note that we can't just extend the integrals further since $f(t)$ is undefined when $t<0$. Instead, we just add it onto all sides.