Let $\alpha(n)= 1-\frac{1}{2}+\frac{1}{3}-\ldots \ldots+(-1)^{n-1} \frac{1}{n}, n \in N$
If $\frac{p_{n}}{q_{n}}=\alpha(n)$
where $p_{n}, q_{n} \in N$ and $h c f\left(p_{n}, q_{n}\right)=1,$ then prove that $p_{n}$ is odd. Also prove or disprove that 1979 divides $p_{1319}$ : (where $a | n: a$ divides $n$ )
My solution: I found manually , till n=6. The numerator is coming odd... but how do i prove it. Also in second part how to start?
Hint 1: Take common denominator. Track what happens to the $\frac{1}{2^k}$ term where $ 2^{k+1} > n \geq 2^k$
Hint 2: Rewrite the series. Observe that $ 1319 \approx \frac{2}{3} \times 1979$.
Also, 1979 is prime.