I was reading some notes on the Dirichlet Kernel and they have a proof of how it reduces to $\sin(2\pi(N+ 1/2)t)/\sin(\pi t)$. I could follow the steps except for one early step which is the following:
$$D_{N}(t) = \sum_{n=-N}^{n=N} e^{ 2 \pi i n t} = e^{-2 \pi i Nt} \sum_{n=0}^{n=2N} e^{2 \pi i n t}$$
The first equals sign is by the definition of the Dircihlet kernel. But I don't understand how the second wequality is obtained. Thanks and I'm sorry for this question. I just hate to skip steps of proofs without understanding them.
This is called index shift. Introduce $m=n+N$. Since $n$ runs from $-N$ to $N$, the new index $m$ runs from $0$ to $2N$. So,
$$\sum_{n=-N}^{n=N} e^{ 2 \pi i n t} = \sum_{m=0}^{m=2N} e^{ 2 \pi i (m-N) t} = \sum_{m=0}^{m=2N} e^{ 2 \pi i m t}e^{ -2 \pi i N t}$$ Here we see the common factor $e^{ -2 \pi i N t}$ and move it outside to get $$e^{ -2 \pi i N t}\sum_{m=0}^{m=2N} e^{ 2 \pi i m t}$$ Lastly, the indices are dummy variables; it does not matter how they are called. So the last formula may just as well be written $$e^{ -2 \pi i N t}\sum_{n=0}^{n=2N} e^{ 2 \pi i n t}$$