
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Investigate the convergence of the alternating series:
1) \( 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+(-1)^{n-1} \frac{1}{n}+\ldots \);
2) \( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots+(-1)^{n-1} \frac{1}{2n-1}+\ldots \);
3) \( \frac{\sin \alpha}{1}+\frac{\sin 2 \alpha}{2^{2}}+\frac{\sin 3 \alpha}{3^{2}}+\ldots \) ( \(\alpha\) is any integer);
4) \( 2-\frac{3}{2}+\frac{4}{3}-\frac{5}{4}+\ldots+(-1)^{n-1} \frac{n+1}{n}+\ldots \).

Present the answer in LaTex format: \boxed{Your answer}
user