Find the sum of the series $$\sum_{k=0}^\infty \frac{(-1)^k}{5^k}$$
I'm wondering if this is divergent since (-1)^k is divergent as per the rules of geometric series where $abs(x) \geq 1$ then $\sum_{k=0}^\infty x^k$ diverges.
Since we know that $$\sum_{k=0}^\infty \frac{(-1)^k}{5^k}=$$
$$\sum_{k=0}^\infty (-1)^k\frac{1}{5^k}$$ and $(-1)^k$ is a divergent then the rest of thte series is divergent? Is that logic right?
Best Answer
This is a geometric series: $$\sum_{n=0}^{\infty}{r^n}=\dfrac{1}{1-r}$$ when $|r|<1$.