Is there an entire function satisfying the following criteria?
No there isn't, since $f$ is holomorphic, we have $ f(x) =_{x \to 0} \sum_{k=0}^{\infty}c_k x^k $, so $f(\frac{1}{n}) = c_0+\frac{c_1}{n} +o(\frac{1}{n})=\frac{(-1)^n}{n} + o(\frac{1}{n})$ so $c_0=0$ and we get $$ c_1 = (-1)^n +o(1) $$ ie. $c_1 - (-1)^n \to_{n \infty} 0$ which is absurd.