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haz lo ejercicios basado en la forma en lo que se lo resuelve en esta imagen

I'll solve the previous exercises based on the style shown in the image you provided. Here's how the solutions would look:

### a) \(\sum_{k=1}^{3} (2 - k)^2\)
Let's calculate each term in the summation:

\[
\sum_{k=1}^{3} (2 - k)^2 = (2-1)^2 + (2-2)^2 + (2-3)^2
\]

Calculate each expression:

\[
1^2 + 0^2 + (-1)^2
\]

Now, compute the squares:

\[
1 + 0 + 1 = 2
\]

So, the answer is \( \boxed{2} \).

### b) \(\sum_{k=0}^{2} 2^k\)
Let's calculate each term in the summation:

\[
\sum_{k=0}^{2} 2^k = 2^0 + 2^1 + 2^2
\]

Calculate each power of 2:

\[
1 + 2 + 4
\]

Now, add the results:

\[
1 + 2 + 4 = 7
\]

So, the answer is \( \boxed{7} \).

### c) \(\sum_{k=0}^{2} (-3)^k\)
Let's calculate each term in the summation:

\[
\sum_{k=0}^{2} (-3)^k = (-3)^0 + (-3)^1 + (-3)^2
\]

Calculate each power:

\[
1 - 3 + 9
\]

Now, sum the results:

\[
1 - 3 + 9 = 7
\]

So, the answer is \( \boxed{7} \).

### d) \(\sum_{k=1}^{4} \frac{1}{k - 2}\)
Let's calculate each term in the summation:

\[
\sum_{k=1}^{4} \frac{1}{k - 2} = \frac{1}{1 - 2} + \frac{1}{2 - 2} + \frac{1}{3 - 2} + \frac{1}{4 - 2}
\]

Now, calculate each term:

\[
\frac{1}{-1} + \frac{1}{0} + \frac{1}{1} + \frac{1}{2}
\]

\[
-1 + \text{undefined} + 1 + \frac{1}{2}
\]

Since the term \(\frac{1}{0}\) is undefined, the sum does not exist as defined. Thus, the result is **undefined**.

### e) \(\sum_{k=1}^{3} (-1)^{k-1}\)
Let's calculate each term in the summation:

\[
\sum_{k=1}^{3} (-1)^{k-1} = (-1)^{1-1} + (-1)^{2-1} + (-1)^{3-1}
\]

Calculate each expression:

\[
1 + (-1) + 1
\]

Now, sum the results:

\[
1 - 1 + 1 = 1
\]

So, the answer is \( \boxed{1} \).

---

Do you have any questions or need further details on any of these problems?

Here are 5 relative questions you might consider:
1. How do you evaluate summations with complex exponents or factorials?
2. What happens if a summation range includes negative or fractional indices?
3. How does changing the bounds of summation affect the result?
4. What are some methods to evaluate summations where the formula inside the summation is complex?
5. How can summations be used to approximate integrals?

**Tip:** When dealing with undefined values in summations, it's important to identify these early to avoid incorrect results.

Learn how to solve exercises involving summation and exponential rules with detailed step-by-step solutions. This problem covers key concepts such as summation formulas, rules for exponents, and handling undefined terms. Suitable for students in Grades 10-12.

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