To determine which of the sequences from the image have a converging subsequence, we can analyze each option:

• a. $\left( -1 + \frac{1}{n} \right)^n$: This sequence behaves similarly to the limit definition of $e^{-1}$. As $n \to \infty$, the expression $\left( -1 + \frac{1}{n} \right)^n$ converges to $e^{-1}$. Since it converges, it has a converging subsequence.

• b. $\tan^{-1}(n)$: The inverse tangent of $n$ approaches $\frac{\pi}{2}$ as $n$ increases. Therefore, the sequence itself converges and hence has a converging subsequence.

• c. $(-n)^n$: This sequence diverges very quickly as the base grows exponentially negative. It does not have a converging subsequence.

• d. Harmonic series $1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$: The harmonic series diverges to infinity as $n$ increases. There is no converging subsequence here.

• e. $\tan(n)$: The tangent function is periodic and unbounded. It does not converge, nor does it have a converging subsequence.

• f. $1 - \frac{1}{2} + \frac{1}{3} - \dots + (-1)^n \frac{1}{n}$: This is an alternating harmonic series, which is known to converge. Therefore, it has a converging subsequence.

• g. $(-1)^n$: This sequence oscillates between -1 and 1, but does not converge. However, subsequences like taking just even-indexed or odd-indexed terms would yield converging subsequences (all 1’s or all -1’s), so it has a converging subsequence.

Final Answer:

• Sequences a, f, and g have converging subsequences.

Would you like a more detailed explanation of any particular sequence?

Here are five follow-up questions:

1. How can we identify converging subsequences in an oscillating sequence?
2. What is the behavior of the harmonic series as $n \to \infty$?
3. What properties of a sequence guarantee that it has a converging subsequence?
4. How does the alternating harmonic series differ from the regular harmonic series in terms of convergence?
5. What are some examples of sequences that do not have converging subsequences?

Tip: If a sequence is bounded and monotonic, it will always converge. If not, it's helpful to examine subsequences for potential convergence!