On page 56 of Tom Apostol's Intro To Analytic Number Theory, he uses the Euler Summation formula (references also in this question) to expand the finite sum.
$$\sum_{n \leq x} \frac{1}{n^s} $$
See image:
In the algebra manipulation he explains how for $s>1$ the expression $C(s) = 1 - \frac{1}{1-s} - s \int_1^\infty \frac{t-[t]}{t^{s+1}}dt$ tends to $\zeta(s)$.
However for $0<s<1$ he also mains the claim, but doesn't explain it.
In my understanding, the key element $\frac{x^{1-s}}{1-s}$ doesn't tend to 0 as $x \rightarrow \infty$.
I have asked a few others and we can't see how $C(s) = \zeta(s)$ for $0<s<1$.
From (7), use algebra to obtain $$ \sum_{x \leq n} \frac{1}{n^s} - \frac{x^{1-s}}{1-s} = C(s) + O(x^{-s}) \text{.} $$ Now take limits as $x \rightarrow \infty$. As Apostol notes, "$x^{-s} \rightarrow 0$", so we obtain \begin{align*} \lim_{x \rightarrow \infty} \left( \sum_{n \leq x} \frac{1}{n^s} - \frac{x^{1-s}}{1-s} \right) &= \lim_{x \rightarrow \infty} \left( C(s) + O(x^{-s}) \right) \\ &= \lim_{x \rightarrow \infty} C(s) + \lim_{x \rightarrow \infty}O(x^{-s}) \\ &= C(s) + 0 \\ &= C(s) \text{.} \end{align*} (That both resulting limits exist justifies distributing the limit over the addition.)
Based on comments, a few words about analytic continuation.
There is a function $\zeta(z)$, Riemann's zeta function that is defined on $\Bbb{C} \smallsetminus \{1\}$. This function has a simple pole at $1$. It is meromorphic, or, what is the same thing, is holomorphic on any domain (open, connected subset of $\Bbb{C}$) excluding $1$. There is a Dirichlet series for $\zeta(z)$ valid for $\Re(z) > 1$ because its abscissa of convergence is $\Re(z) = 1$. (The pole prevents extension of the region of convergence of this series onto and to the left of this abscissa in a manner analogous to a pole preventing a larger radius of convergence for a power series.) So, we have $$ \zeta(z) = \sum_{n=1}^\infty \frac{1}{n^z} , \Re(z) > 1 \text{.} $$ But $\zeta$ is defined on the rest of the complex plane (excluding $1$). How does this happen?
Apostol is only interested is real $z$, and he finds another representation of $\zeta(s)$ for the strip $0 < s < 1$ by starting with the Dirichlet series, showing that it is identical to another meromorphic function, $C(s)$, on the half-line $s > 1$ and then observing that this new representation also converges on the strip $0 < s < 1$. Then, using the identity theorem, $C(s)$ is the holomorphic (a.k.a. analytic) continuation of the Dirichlet series to the interval $0 < s < 1$.
In the background, this work is actually happening on $\Bbb{C}$. $\zeta(z)$ is the continuation of the Dirichlet series. The Dirichlet series converges on $\Re(z) > 1$. Apostol's rewrite gives an expression, $C(z)$, that converges on $\Re(z) > 0$ and $z \neq 1$. Using the identity theorem, $C(z)$ is the unique extension of $\zeta(z)$ from the half-plane $\Re(z) > 1$ to the (almost) half-plane $\Re(z) > 0$ and $z \neq 1$.