To analyze the series $\sum_{n=0}^\infty \frac{1}{2} \cdot 1^n$:

Series Expression

$\sum_{n=0}^\infty \frac{1}{2} \cdot 1^n$

This simplifies to: $\sum_{n=0}^\infty \frac{1}{2}$ since $1^n = 1$ for all $n$.

Behavior of the Series

The series becomes: $\sum_{n=0}^\infty \frac{1}{2} = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots$

This is an infinite series where each term is constant ($\frac{1}{2}$). Such a series does not converge because the partial sums grow indefinitely: S_N = \frac{1}{2}N \quad \text{(where N is the number of terms in the partial sum)}.

Conclusion

The series diverges because the terms do not decrease and the partial sums approach infinity.

Would you like me to explain this in further detail or analyze a related case?

Here are 5 related questions for exploration:

1. What happens if the terms of the series decrease geometrically (e.g., $\frac{1}{2^n}$) instead of staying constant?
2. How can you determine if a series converges or diverges in general?
3. What is the role of the ratio test in evaluating series convergence?
4. How does the divergence test apply to this series specifically?
5. What are some examples of conditionally convergent series?

Tip: Always check the behavior of the general term of a series as $n \to \infty$ to get an idea about convergence or divergence.