The problem involves two infinite series:

A)
$\sum_{n=1}^\infty \frac{n + 2}{5n + 1},$

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

Let’s analyze both series and determine their convergence or divergence.

Analysis of A)

$a_n = \frac{n + 2}{5n + 1}.$

Asymptotic Behavior of $a_n$:

For large $n$, the dominant terms in the numerator and denominator are $n$ and $5n$, respectively: $a_n \sim \frac{n}{5n} = \frac{1}{5}.$

Since $a_n \nrightarrow 0$ as $n \to \infty$ (it approaches $\frac{1}{5}$), the series diverges by the test for divergence.

Analysis of B)

$b_n = \frac{1}{2^n}.$

This is a geometric series with a common ratio $r = \frac{1}{2}$, where $0 < r < 1$. The formula for the sum of an infinite geometric series is: $S = \frac{a}{1 - r},$ where $a$ is the first term.

Here, $a = \frac{1}{2^1} = \frac{1}{2}$ and $r = \frac{1}{2}$. Thus: $S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1.$

The series converges to $1$.

Final Conclusion:

• Series A diverges.
• Series B converges.

Would you like further details or help solving related questions?

Related Questions:

1. How do you formally prove the divergence of a series?
2. What is the divergence test, and when is it applicable?
3. How can you generalize the formula for geometric series sums?
4. What happens if the denominator of series A is modified?
5. How does the rate of growth of the numerator and denominator affect convergence?

Tip:

When testing for convergence, always check if the terms of the series approach zero. If not, the series diverges immediately.