Determine if $\sum_{t=0}^\infty \frac{(-1)^{t^2}t^2}{4+t^2}$ is absolutely convergent, conditionally convergent, or divergent

Question

Determine if $$\sum_{t=0}^\infty \frac{(-1)^{t^2}t^2}{4+t^2}$$ is absolutely convergent, conditionally convergent, or divergent. I have tried the Ratio test, but the limit comes out to be $1$, which means that it is inconclusive for both the Ratio and Root test. What can I do now?

Answer 1

Observe that

$$\frac{t^2}{4+t^2}\xrightarrow[t\to\infty]{}1\implies\frac{(-1)^{t^2}t^2}{4+t^2}\rlap{\;\;\;\;/}\xrightarrow[t\to\infty]{}0$$