Evaluate Sum: ∑(k=2 To 5) (-2/k!) - Step-by-Step

Hey guys! Today, we're diving into a fun math problem: evaluating the sum of a series. Specifically, we're tackling the series ∑(k=2 to 5) (-2/k!). This might look a bit intimidating at first, but don't worry, we'll break it down step-by-step so it's super easy to understand. So, grab your calculators (or your brains, if you're feeling extra sharp!), and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the problem is asking. The expression ∑(k=2 to 5) (-2/k!) is a summation notation. Let's dissect it:

• ∑ (Sigma): This is the summation symbol, which means we're going to add up a series of terms.
• k=2: This tells us where to start our summation. We'll begin with k equal to 2.
• 5: This is where we end our summation. We'll stop when k reaches 5.
• (-2/k!): This is the formula for each term in the series. We'll plug in the value of k, calculate the result, and then add it to the running total.
• k! (k factorial): This means k factorial, which is the product of all positive integers up to k. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120.

In plain English, we need to calculate the value of (-2/k!) for k = 2, 3, 4, and 5, and then add those values together. Easy peasy, right? Let's move on to the step-by-step calculation.

Step 1: Calculate the Factorials

The first thing we need to do is calculate the factorials for each value of k from 2 to 5. Remember, the factorial of a number is the product of all positive integers less than or equal to that number. This step is crucial because it forms the denominator of our fractions, significantly influencing the final sum. Getting these right is key to solving the problem accurately. Here’s how we calculate them:

• 2! = 2 * 1 = 2
• 3! = 3 * 2 * 1 = 6
• 4! = 4 * 3 * 2 * 1 = 24
• 5! = 5 * 4 * 3 * 2 * 1 = 120

Now that we have our factorials, we can move on to the next step: calculating each term in the series. By accurately calculating these factorials, we set a solid foundation for the rest of the computation. Each factorial represents the number of ways to arrange 'k' distinct items, a concept that extends beyond simple calculation into combinatorics and probability. These values will be plugged into the expression (-2/k!) to find each term of the series, which we will then sum up to find our final answer. It’s all about breaking down a complex problem into manageable parts, and this is our first big step!

Step 2: Calculate Each Term

Now that we've got the factorials sorted out, let's calculate each term in the series. Remember, the formula for each term is (-2/k!). We'll simply plug in the values of k (2, 3, 4, and 5) and calculate the result. This part is where we combine the factorials we just computed with the constant -2, giving us the individual components that we'll sum up in the next step. It's a direct application of the formula and a great way to see how each term contributes to the overall sum.

• For k = 2: (-2/2!) = (-2/2) = -1
• For k = 3: (-2/3!) = (-2/6) = -1/3
• For k = 4: (-2/4!) = (-2/24) = -1/12
• For k = 5: (-2/5!) = (-2/120) = -1/60

So, we now have all the individual terms of our series. Each fraction represents a slice of the total sum, and you can already start to get a sense of how they'll add up. Notice how the denominators get larger as k increases, which means the terms get smaller in magnitude. This is a characteristic of many series involving factorials, which often converge quickly. With these terms in hand, we're just one step away from finding the final answer. Next, we’ll add them all together!

Step 3: Sum the Terms

Alright, we're in the home stretch! We've calculated all the individual terms, and now it's time to add them up. This is the final step in evaluating the sum of the series. We'll take the values we calculated in the previous step and simply add them together. This step is like the grand finale, bringing all our previous work together to get the final result. Let's make sure we add these fractions correctly to nail the answer.

The sum is: -1 + (-1/3) + (-1/12) + (-1/60). To add these fractions, we need a common denominator. The least common multiple of 3, 12, and 60 is 60, so let's convert each fraction:

• -1 = -60/60
• -1/3 = -20/60
• -1/12 = -5/60
• -1/60 = -1/60

Now we can add the fractions: (-60/60) + (-20/60) + (-5/60) + (-1/60) = -86/60. We can simplify this fraction by dividing both numerator and denominator by their greatest common divisor, which is 2:

-86/60 = -43/30

So, the sum of the series is -43/30. We’ve done it! By adding these terms, we've arrived at our final answer. This process shows how individual pieces come together to form a whole, a concept that’s not just applicable in math, but in many aspects of life. The result, -43/30, is a single number that represents the accumulation of all the terms in the series. Next time you see a summation, remember this step-by-step approach, and you'll be able to tackle it with confidence.

Final Answer

Therefore, the sum of the series ∑(k=2 to 5) (-2/k!) is -43/30. We've successfully evaluated the sum by breaking it down into smaller, manageable steps. We started by understanding the summation notation, then calculated the factorials, found each term, and finally, added them all up. This systematic approach is a powerful tool for solving mathematical problems, and it's something you can apply to all sorts of challenges.

Key Takeaways:

• Understanding the Notation: Make sure you understand what the summation notation means before you start calculating.
• Breaking it Down: Complex problems become easier when you break them into smaller steps.
• Factorials are Your Friends: Don't be intimidated by factorials; they're just a product of numbers.
• Common Denominators are Key: When adding fractions, always find a common denominator.

Why is this Important?

You might be wondering,