Problem 39 - Calculus Early Transcendentals

Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.) a. \(1+2+3+4+5\) b. \(4+5+6+7+8+9\) c. \(1^{2}+2^{2}+3^{2}+4^{2}\) d. \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\)

Step-by-Step Solution

Answer

Question: Express the following sums using sigma notation: a. 1+2+3+4+5 b. 4+5+6+7+8+9 c. \(1^{2}+2^{2}+3^{2}+4^{2}\) d. 1+\(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\) Answer: a. \(\sum_{n=1}^{5} n\) b. \(\sum_{n=1}^{6} (n+3)\) c. \(\sum_{n=1}^{4} n^2\) d. \(\sum_{n=1}^{4} \frac{1}{n}\)

Step 1: Identify the formula for the terms in the sequence

We can see that each term of the sequence is a natural number that increases by 1 for each subsequent term. The general formula for this sequence is n, where n is the position of the term in the sequence.

Step 2: Determine the starting and ending terms of the sequence

The first term of the sequence is 1, and the last term is 5.

Step 3: Write the sum in sigma notation

Using the information we have gathered, we can write the sum in sigma notation as: \(\sum_{n=1}^{5} n\) b. \(4+5+6+7+8+9\)

Step 1: Identify the formula for the terms in the sequence

Each term of the sequence is one more than the previous term. The general formula for this sequence is n + 3, where n is the position of the term in the sequence.

Step 2: Determine the starting and ending terms of the sequence

The first term of the sequence is 4 (when n=1), and the last term is 9.

Step 3: Write the sum in sigma notation

Using the information we have gathered, we can write the sum in sigma notation as: \(\sum_{n=1}^{6} (n+3)\) c. \(1^{2}+2^{2}+3^{2}+4^{2}\)

Step 1: Identify the formula for the terms in the sequence

We can see that the sequence contains the square of each natural number. The general formula for this sequence is \(n^2\), where n is the position of the term in the sequence.

Step 2: Determine the starting and ending terms of the sequence

The first term of the sequence is \(1^2\), and the last term is \(4^2\).

Step 3: Write the sum in sigma notation

Using the information we have gathered, we can write the sum in sigma notation as: \(\sum_{n=1}^{4} n^2\) d. \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\)

Step 1: Identify the formula for the terms in the sequence

The sequence contains the reciprocal of each natural number. The general formula for this sequence is \(\frac{1}{n}\), where n is the position of the term in the sequence.

Step 2: Determine the starting and ending terms of the sequence

The first term of the sequence is 1 (when n=1), and the last term is \(\frac{1}{4}\) (when n=4).

Step 3: Write the sum in sigma notation

Using the information we have gathered, we can write the sum in sigma notation as: \(\sum_{n=1}^{4} \frac{1}{n}\)