First of all let me say that I am a physicist and therefore it is sometimes hard for me to understand some mathematical steps... Now, I've been trying to obtain the well known result for the zeta function $\zeta(-1)=-\frac{1}{12}$, but my string theory teacher is asking the students to show this in a very specific way. To do so, we consider the gamma function: $$\Gamma(s)=\int_0^\infty dt t^{s-1}e^{-t}$$ and the zeta function: $$\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$$ with $s$ a complex number with real part greater than 1. The firststep of the demonstration is to show that using a transformation $t \rightarrow nt$ we can write $$\Gamma(s)\zeta(s)=\int_0^\infty dt \frac{t^{s-1}}{e^t-1}$$. I had no problems here, I just wrote the definition of a geometric series of argument $e^-t$ to obtain the result. Then, the second step consisted on showing that we can also write $$\Gamma(s)\zeta(s)=\int_0^1 dt\left(\frac{1}{e^t-1}-\frac{1}{t}+\frac{1}{2}-\frac{t}{12}\right)+\int_1^\infty dt \left(\frac{t^{s-1}}{e^t-1}\right)+\frac{1}{s-1}-\frac{1}{2s}+\frac{1}{12(s+1)}.$$ I also did not have any problems here because the integrals of the parcels inside the first brackets cancel the terms depending on $s$ outside the integrals and so it was just a matter of verifying that this result was the same as before.
Now, this is where I am stuck: we started by saying that $s$ is a complex number with real part greater than 1. Now, we are supposed to explain why this last result i wrote is in fact well-behaved for any complex $s$ with real part greater than $-2$, and I don't understand how is that possible since we have terms that diverge for $s=\{-1,0,1\}$.
Finally, the last step of the demonstration consists of using the previous result, that is well-behaved for $s$ with real part greater than $-2$, and also the fact that the gamma function as pole at $s=-1$ with residue $-1$ to conclude that $\zeta(-1)=-\frac{1}{12}$. I am also not able to conclude anything because I really don't know how. A friend of mine said he did something using the Bernoulli numbers, but I am not sure about what he did... Can any one help me here?