How to prove the sums have not the same sign.

Question

Let $(a_n)_{n\ge 0}$ be a non periodic sequence of elements in $\{0,1\}$.

Could we prove that The sums $$\sum_{i=0}^n(-1)^{a_i}$$ and $$\sum_{i=0}^n(-1)^{i+a_i}$$

have not necessarily the same sign at least for large enough $n$.

Answer 1

If the $a_i$'s are mostly $0$'s with an occasional $1$, then the first sum will diverge to $+\infty$. If the occasional $1$'s all appear at even values of $i$, then the second sum will diverge to $-\infty$.

On the other hand, if the occasional $1$'s all appear at odd values of $i$, the second sum will also diverge to $+\infty$. In general, by modulating the parity of where the $1$'s occasionally appear, you can get the second sum to do pretty much whatever you want.