Question

Geometric series: For each of the series that follow, find the sum or explain why the series diverges.

$\sum _{k=0}^{\infty }{\left(1\right)}^k$

Short answer

The given series diverges.

Step-by-step solution

Step 1. Given Information

The given series is$\sum _{k=0}^{\infty }{\left(1\right)}^k$.

Step 2. Determine if a geometric series  

• Find the ratio of the consecutive terms.

$\begin{array}{rcl}\frac{{\left(1\right)}^{k+1}}{{\left(1\right)}^k}& =& 1^{k+1-k}\\ & =& 1\end{array}$

• Since the ratio of the consecutive terms is same, the given series is a geometric series with a common ratio $r=1$.

Step 3. Determine the convergence  

• Determine the first term and the common ratio.

$\begin{array}{rcl}c& =& {\left(1\right)}^0\\ & =& 1\\ r& =& 1\end{array}$

• If $\left|r\right|<1$, the series converges.
• So, the given series diverges.