Question

In Problems 21-24, for each of the four series (a)-(d), identify which have the same number of terms and which have the same value. (a) \(\sum_{i=1}^{5} i^{2}\) (b) \(\sum_{j=0}^{4}(j+1)^{2}\) (c) \(5+4+\ldots+0\) (d) \(5^{2}+4^{2}+\ldots+1^{2}\)

Short answer

Answer: Series (a), (b), and (d) have the same number of terms and the same value.

Step-by-step solution

Simplify the series if needed

In this case, we need to simplify (b) and (c) to make them consistent with the other series. (b) \(\sum_{j=0}^{4}(j+1)^{2}\) can be simplified as \(\sum_{j=1}^{5} j^{2}\), since replacing \(j\) by \(j+1\) gives the same sum. (c) \(5+4+\ldots+0\) can be rewritten as \(\sum_{i=0}^{5} i\) by using the series notation. Now we have the following series: (a) \(\sum_{i=1}^{5} i^{2}\) (b) \(\sum_{j=1}^{5} j^{2}\) (c) \(\sum_{i=0}^{5} i\) (d) \(5^{2}+4^{2}+\ldots+1^{2}\)

Compute the number of terms in each series

(a) The series goes from \(i=1\) to \(i=5\), so there are \(5 - 1 + 1 = 5\) terms. (b) The series goes from \(j=1\) to \(j=5\), so there are \(5 - 1 + 1 = 5\) terms. (c) The series goes from \(i=0\) to \(i=5\), so there are \(5 - 0 + 1 = 6\) terms. (d) From the simplified series notation, the series is the same as (a) and (b), so there are 5 terms. Comparing the number of terms, (a), (b), and (d) have the same number of terms (5 terms), while (c) has a different number of terms (6 terms).

Calculate the value of each series

(a) The value is \(1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} = 1 + 4 + 9 + 16 + 25 = 55\) (b) The value of this simplified series is the same as (a), which is 55. (c) The value is \(0 + 1 + 2 + 3 + 4 + 5 = 15\) (d) The value is the same as (a) and (b), which is 55. Comparing the values, (a), (b), and (d) have the same value (55), while (c) has a different value (15). To summarize, series (a), (b), and (d) have the same number of terms and the same value. Series (c) has a different number of terms and a different value.

Key concepts

Sum notation

Sum notation is a mathematical shorthand to denote the sum of a sequence of terms. It uses the Greek letter sigma (\( \Sigma \)) to represent the operation of summation. For example, the expression \( \sum_{i=1}^{5} i^{2} \) indicates that you should sum the square of each integer starting from 1 up to 5.
In our example, the sum notation in \((a)\), \((b)\), and \((c)\) clearly specifies the range and the formula for each term. Understanding sum notation is critical because it compacts a potentially long addition process into a concise expression. It clarifies how each term in the series is derived and the bounds of the series.
Summation helps in simplifying the representation of complex series, especially in cases involving a large number of terms, minimizing the chances of arithmetic errors.

Number of terms

Determining the number of terms in a series is a vital step in understanding and comparing series. The number of terms refers to how many terms are added together in a given series.
In step 2 of our example, we calculate the number of terms for each series:

• Series (a): Term count is \(5\) because it runs from \(i=1\) to \(i=5\)
• Series (b): Similarly has \(5\) terms as it also goes from \(j=1\) to \(j=5\)
• Series (c): This has \(6\) terms running from \(i=0\) to \(i=5\)
• Series (d): Given directly, aligns with \(5\) terms

Determining the number of terms allows you to compare different series flexibly and identify sequences with similar characteristics. It's important because it also affects calculation methods and results.

Value of series

The value of a series is the final result you get after summing all its terms. Calculating this value gives insight into the sum's magnitude and allows for comparison between different series.
The value of each series example calculated in step 3 is:

• Series (a) and (b): Both add up to \(55\)
• Series (c): Adds up to \(15\), distinct from the others
• Series (d): Matches series \((a)\) and \((b)\) with a sum of \(55\)

Understanding the value helps validate the structure set by the sum notation and term count. It confirms whether simplifications or assumptions maintain the same ultimate results.

Simplification of series

Series simplification is the process of transforming a series into an easier or more intuitive expression. In some cases, two different series might express ultimately the same sum.
Simplifying a series can involve rewriting the terms in a way that retains the value and the number of terms but presents them more consistently.
In this exercise, series (b) was rewritten from \(\sum_{j=0}^{4}(j+1)^{2}\) to \(\sum_{j=1}^{5} j^{2}\), aligning with series \((a)\). This transformation retained the original sum and term count while making the comparison easier.
Understanding and applying simplification techniques allow us to find equivalencies between series and make complex calculations more manageable. It ensures that while differences in notation may arise, the essential mathematical relationships remain unchanged.