Problem 57 - Algebra 2

Question

Evaluate the sum \(\sum_{n=1}^{3}\left(\frac{1}{n+1}\right)^{2} .\) Enter your answer as a decimal to the nearest hundredth.

Step-by-Step Solution

Answer

The evaluated sum of the series to the nearest hundredth is 0.42.

Step 1: Substitute n=1

Start by substituting n=1 into the series. This makes the first term of the series to be \(\left(\frac{1}{1+1}\right)^{2} = \left(\frac{1}{2}\right)^{2} = 0.25\).

Step 2: Substitute n=2

Next substitute n=2 into the series. This makes the second term of the series to be \(\left(\frac{1}{2+1}\right)^{2} =\left(\frac{1}{3}\right)^{2} = 0.11\).

Step 3: Substitute n=3

Finally, substitute n=3 into the series. This makes the third term of the series to be \(\left(\frac{1}{3+1}\right)^{2} = \left(\frac{1}{4}\right)^{2} = 0.0625\).

Step 4: Calculate the Sum

Now add all three calculated terms together to get the sum of the series. So, the sum is \(0.25+0.11+0.0625 = 0.4225\)

Key Concepts

Series Evaluation

The concept of series evaluation involves finding the collective sum of a sequence of terms. Here, we are given a specific type of mathematical series known as a summation series. This series is expressed as \(\sum_{n=1}^{3}\left(\frac{1}{n+1}\right)^{2}\). This notation signifies that we are summing multiple terms that are determined by plugging different integer values into the expression \(\left(\frac{1}{n+1}\right)^{2}\).

To evaluate this series:

• Identify the range of \(n\), which in this case is from 1 to 3.
• Substitute each integer value sequentially for \(n\) into the expression.
• Calculate the resulting value for each of these terms.
• Add all these evaluated terms to find the total sum.

By following this method, we combine each distinct term's contribution into one overall sum.

Decimal Approximation

Decimal approximation simplifies numbers into a more digestible form, typically useful in contexts where exact precision is either unnecessary or overly complex to use. When dealing with series evaluation, the results are often rounded to a certain decimal place for simplification. In this exercise, the sum of the series needs to be expressed as a decimal to the nearest hundredth.

Rounding involves looking at the number immediately after the desired decimal place:

• If it is 5 or higher, round up the last desired decimal place by one.
• If it is less than 5, keep the rounding decimal unchanged.

In the given series, each evaluated term was expressed in decimal form before adding. The final summation gave us \(0.4225\), which is rounded to \(0.42\) when expressed to the nearest hundredth. This level of approximation aids in quick interpretation and application of results.

Sequence Substitution

Sequence substitution is a step used to individually compute each term of the series by replacing the variable in the given expression with specific values from the defined sequence. In our problem, we substitute the values 1, 2, and 3 for \(n\) in the expression \(\left(\frac{1}{n+1}\right)^{2}\).

To perform sequence substitution:

• Start with the smallest value in the sequence, substituting it into the expression.
• Compute the result for that substitution.
• Proceed to the next value in the sequence, continuing until all specified numbers have been used.

For instance, substituting \(n = 1\), we calculate \(\left(\frac{1}{2}\right)^{2} = 0.25\). Likewise, \(n = 2\) results in \(\left(\frac{1}{3}\right)^{2} = 0.11\), and \(n = 3\) results in \(\left(\frac{1}{4}\right)^{2} = 0.0625\). This systematic approach breaks the series down into manageable parts, significantly easing the evaluation process.