Let's analyze the statements and determine which ones are true:

1. Statement 1:
   $a_k > \frac{1}{k} \text{ for all } k \in \mathbb{N}, \text{ so } \sum_{k=1}^{\infty} a_k \text{ diverges.}$
   This statement is true. If each term $a_k$ is greater than $\frac{1}{k}$ and we know that the harmonic series $\sum \frac{1}{k}$ diverges, then by comparison, $\sum a_k$ must also diverge.

2. Statement 2:
   $a_k > |b_k| \text{ for all } k \in \mathbb{N} \text{ and } \sum_{k=1}^{\infty} |b_k| \text{ diverges, so } \sum_{k=1}^{\infty} a_k \text{ diverges.}$
   This is also true. If $a_k > |b_k|$ and $\sum |b_k|$ diverges, then $\sum a_k$ will diverge as well, by comparison.

3. Statement 3:
   $\sum_{k=1}^{\infty} a_k = a \text{ implies } \lim_{k \to \infty} a_k = a.$
   This is false. If a series converges, it does not imply that the terms of the sequence converge to the sum. For instance, a convergent series has terms tending to zero as $k \to \infty$, not necessarily to $a$.

4. Statement 4:
   $|a_k| < |b_k| \text{ for all } k \in \mathbb{N}, \text{ and } \sum_{k=1}^{\infty} a_k \text{ converges, so } \sum_{k=1}^{\infty} b_k \text{ converges.}$
   This statement is false. The convergence of $\sum a_k$ does not imply the convergence of $\sum b_k$ even if $|a_k| < |b_k|$. The comparison test would require $|b_k| < |a_k|$ if we wanted $b_k$ to converge based on $a_k$.

5. Statement 5:
   $\lim_{k \to \infty} a_k = 0 \text{ implies that } \sum_{k=1}^{\infty} a_k \text{ converges.}$
   This is false. The fact that $a_k \to 0$ as $k \to \infty$ is a necessary condition for convergence but not sufficient. For example, the harmonic series $\sum \frac{1}{k}$ diverges even though $\frac{1}{k} \to 0$.

6. Statement 6:
   $|a_k| < \frac{1}{k} \text{ for all } k \in \mathbb{N}, \text{ so } \sum_{k=1}^{\infty} a_k \text{ converges.}$
   This statement is true. If $|a_k| < \frac{1}{k}$ and the harmonic series $\sum \frac{1}{k}$ diverges, we can conclude by the comparison test that $\sum a_k$ converges.

7. Statement 7:
   $\lim_{k \to \infty} a_k = a \text{ with } a \neq 0 \text{ implies } \sum_{k=1}^{\infty} a_k \text{ diverges.}$
   This is true. If $\lim_{k \to \infty} a_k = a \neq 0$, then the series $\sum a_k$ diverges because the terms do not tend to zero.

Conclusion

The statements that are true (marked as checked in the image) are:

• Statement 1
• Statement 2
• Statement 6
• Statement 7

Would you like further details on any of these explanations?

Related Questions

1. Why does the harmonic series diverge?
2. What is the difference between convergence of terms and convergence of a series?
3. How does the comparison test work in proving convergence or divergence?
4. Can a series with terms approaching zero still diverge? Explain with examples.
5. What are the necessary and sufficient conditions for the convergence of an infinite series?

Tip

Always remember that for a series to converge, the terms must approach zero, but this alone does not guarantee convergence; additional conditions like comparison or ratio tests may be required.