The series provided in the image is:

$\sum_{n=1}^{\infty} \frac{1}{2^n - n}.$

Let’s analyze the convergence of this series.

Key Considerations:

1. Convergence Test: To check if a series converges, common methods include:

   • Comparing with a known convergent/divergent series (Comparison Test).
   • Checking if the general term approaches zero (necessary condition for convergence).
   • The Ratio Test or Root Test.

2. Growth of the Denominator: The denominator $2^n - n$ grows exponentially due to the $2^n$ term. This suggests that the terms decrease very quickly.

Steps:

(a) Behavior of the General Term:

The general term is: $a_n = \frac{1}{2^n - n}.$

As $n \to \infty$, $2^n$ dominates over $n$, so: $2^n - n \sim 2^n.$ Thus: $a_n \sim \frac{1}{2^n}.$

(b) Comparison with a Geometric Series:

The series $\sum_{n=1}^\infty \frac{1}{2^n}$ is a convergent geometric series with a common ratio $\frac{1}{2}$. Since $\frac{1}{2^n - n} < \frac{1}{2^n}$ for all $n \geq 2$, the Comparison Test implies that the given series converges.

(c) Final Statement:

The series $\sum_{n=1}^\infty \frac{1}{2^n - n}$ converges due to the rapid growth of the denominator.

Let me know if you'd like more details or if you have questions!

Expanding Questions:

1. How does the Comparison Test work for proving series convergence?
2. What is the necessary condition for a series to converge?
3. Can the Ratio Test be used to confirm the convergence here?
4. How does the growth rate of $2^n$ influence the convergence of the series?
5. What are examples of divergent series with slower-growing denominators?

Tip:

Always simplify the general term of a series to its dominant growth behavior to decide on an appropriate convergence test.