Evaluate the following infinite series or state that the series diverges.

Question

From my textbook.

$$\sum\limits_{k=0}^\infty (-\frac{1}{5})^k$$

My work:

So a constant greater than or equal to $1$ raised to ∞ is ∞.

A number $n$ for $0<n<1$ is $0$. So when taking the limit of this series you get 0 but when formatting the problem a different way $(-1)^k/(5^k)$ it seems like an alternating series. Can someone help me figure this out?

Answer 1

For $-1 <x< 1$, we know $\sum_{i=0}^{\infty}x^i=\frac{1}{1-x}$

So in your problem, x = -1/5, sum is 5/6

Answer 2

You have to solve this by using the geometric series test, where |r|< 1, the series converges, and if |r|> 1, it diverges. Since r= -1/5, and you take the absolute value to get 1/5, the series converges because 1/5 < 1.