Let $S=1-1/3+1/5-1/7+\cdots$. As each term in the series is decreasing and tends to $0$, it is known that their sum exists and is finite by alternating series test. And by considering $\int_0^11/(1+x^2)dx$, it is known that $S=\pi/4$.
I am wondering is there a fast way to see that $S\neq 0$? In general, is it true that for $T=\sum_{n=1}^\infty (-1)^n a_n$, where $a_n>0$ are strictly decreasing and tend to $0$, $T\neq 0$?