https://r2.erweima.ai/mathbot/image/1726776386752leje1iy3.jpg

Let's break down the two sums shown in the image:

### First Series:
The first series is:
\[
3 + 3\left( -\frac{1}{4} \right) + 3\left( -\frac{1}{4} \right)^2 + \cdots
\]
This is a geometric series with the first term \( a = 3 \) and common ratio \( r = -\frac{1}{4} \).

For a geometric series \( a + ar + ar^2 + \cdots \), the sum is given by:
\[
S = \frac{a}{1 - r} \quad \text{if} \quad |r| < 1.
\]
Here, \( a = 3 \) and \( r = -\frac{1}{4} \), and since \( |r| < 1 \), the series converges. Applying the formula:
\[
S = \frac{3}{1 - \left( -\frac{1}{4} \right)} = \frac{3}{1 + \frac{1}{4}} = \frac{3}{\frac{5}{4}} = 3 \times \frac{4}{5} = \frac{12}{5}.
\]
Thus, the sum of the first series is:
\[
\frac{12}{5}.
\]

### Second Series:
The second series is:
\[
\sum_{i=1}^{\infty} 4 \left( \frac{4}{7} \right)^i
\]
This is another geometric series with the first term \( a = 4 \left( \frac{4}{7} \right) = \frac{16}{7} \) and common ratio \( r = \frac{4}{7} \).

Using the sum formula for an infinite geometric series:
\[
S = \frac{a}{1 - r},
\]
we have \( a = \frac{16}{7} \) and \( r = \frac{4}{7} \), so:
\[
S = \frac{\frac{16}{7}}{1 - \frac{4}{7}} = \frac{\frac{16}{7}}{\frac{3}{7}} = \frac{16}{3}.
\]
Thus, the sum of the second series is:
\[
\frac{16}{3}.
\]

### Final Answers:
- First sum: \( \frac{12}{5} \)
- Second sum: \( \frac{16}{3} \)

Would you like further details on these calculations, or do you have any questions?

Here are 5 related questions for you to explore:
1. What is the condition for a geometric series to converge?
2. How can the formula for the sum of a geometric series be derived?
3. What happens if the common ratio \( r \) is greater than or equal to 1 in an infinite series?
4. Can geometric series be used to model real-world phenomena?
5. How does the value of the first term affect the sum of a geometric series?

**Tip:** Always check the absolute value of the common ratio to ensure a geometric series converges!

Calcular cada suma a continuación. Dar valores exactos, no aproximaciones decimales. Si la suma no existe, haga clic en 'No tiene suma'.

Learn how to calculate geometric series with exact sums. The problem involves applying the geometric series sum formula to determine whether the series converges and calculating the exact values.

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