Let's analyze and compute each given sum.

(a) Geometric Series Calculation

The given sum is:

$5 + 5\left(-\frac{3}{4}\right) + 5\left(-\frac{3}{4}\right)^2 + \dots$

This is an infinite geometric series with:

• First term: $a = 5$
• Common ratio: $r = -\frac{3}{4}$

Since the absolute value of the common ratio satisfies $|r| < 1$, the sum of an infinite geometric series is given by:

$S = \frac{a}{1 - r}$

Substituting the values:

$S = \frac{5}{1 - \left(-\frac{3}{4}\right)}$

$S = \frac{5}{1 + \frac{3}{4}}$

$S = \frac{5}{\frac{4}{4} + \frac{3}{4}}$

$S = \frac{5}{\frac{7}{4}}$

$S = 5 \times \frac{4}{7} = \frac{20}{7}$

Thus, the exact sum is:

$\frac{20}{7}$

---

(b) Divergence of the Given Series

The given summation is:

$\sum_{i=1}^{\infty} 4(-4)^i$

This represents an infinite geometric series with:

• First term: $a = 4(-4)^1 = -16$
• Common ratio: $r = -4$

Since $|r| > 1$, the series does not converge, meaning the sum does not exist.

Thus, the correct answer for this part is:

$\text{No sum}$

---

Summary of Answers:

(a) $\frac{20}{7}$
(b) No sum (Divergent series)

Would you like a more detailed explanation on any part?

---

Related Questions:

1. How do you determine if a geometric series converges or diverges?
2. What is the general formula for the sum of an infinite geometric series?
3. What happens if $|r| = 1$ in a geometric series?
4. Can a divergent series still be assigned a sum using advanced techniques?
5. How does the sum formula change for a finite geometric series?

Tip:

For an infinite geometric series to converge, the absolute value of the common ratio must be strictly less than 1, i.e., $|r| < 1$.