Problem 41 - Calculus Early Transcendentals

Question

Sigma notation Evaluate the following expressions. a. \(\sum_{k=1}^{10} k\) b. \(\sum_{k=1}^{6}(2 k+1)\) c. \(\sum_{k=1}^{4} k^{2}\) d. \(\sum_{n=1}^{5}\left(1+n^{2}\right)\) e. \(\sum_{m=1}^{3} \frac{2 m+2}{3} \) f. \(\sum_{j=1}^{3}(3 j-4)\) g. \(\sum_{p=1}^{5}\left(2 p+p^{2}\right)\) h. \(\sum_{n=0}^{4} \sin \frac{n \pi}{2}\)

Step-by-Step Solution

Answer

Question: Evaluate the following expressions in Sigma notation: a) \(\sum_{k=1}^{10} k\) b) \(\sum_{k=1}^{6}(2 k+1)\) c) \(\sum_{k=1}^{4} k^{2}\) d) \(\sum_{n=1}^{5}\left(1+n^{2}\right)\) e) \(\sum_{m=1}^{3} \frac{2 m+2}{3}\) f) \(\sum_{j=1}^{3}(3 j-4)\) g) \(\sum_{p=1}^{5}\left(2 p+p^{2}\right)\) h) \(\sum_{n=0}^{4} \sin \frac{n \pi}{2}\) Answer: a) 55 b) 42 c) 30 d) 60 e) 6 f) 6 g) 115 h) 0

Step 1: Write down the expression with each term

For a: \(\sum_{k=1}^{10} k\) , we need to replace k with each value from 1 to 10 and write down each term: \(1+2+3+\cdots+10\)

Step 2: Calculate the sum

Calculate the sum of the terms: \(1+2+3+\cdots+10 = 55\) #b. Evaluating the expression#

Step 1: Write down the expression with each term

For b: \(\sum_{k=1}^{6}(2 k+1)\) , replace k with 1 to 6 and write down each term: \(3+5+7+\cdots+13\)

Step 2: Calculate the sum

Calculate the sum of the terms: \(3+5+7+\cdots+13 = 42\) #c. Evaluating the expression#

Step 1: Write down the expression with each term

For c: \(\sum_{k=1}^{4} k^{2}\) , replace k with 1 to 4 and write down each term: \(1^{2}+2^{2}+3^{2}+4^{2}\)

Step 2: Calculate the sum

Calculate the sum of the terms: \(1^{2}+2^{2}+3^{2}+4^{2} = 30\) #d. Evaluating the expression#

Step 1: Write down the expression with each term

For d: \(\sum_{n=1}^{5}\left(1+n^{2}\right)\) , replace n with 1 to 5 and write down each term: \(2+5+10+\cdots+26\)

Step 2: Calculate the sum

Calculate the sum of the terms: \(2+5+10+\cdots+26 = 60\) #e. Evaluating the expression#

Step 1: Write down the expression with each term

For e: \(\sum_{m=1}^{3} \frac{2 m+2}{3}\) , replace m with 1 to 3 and write down each term: \(\frac{4}{3}+\frac{6}{3}+\frac{8}{3}\)

Step 2: Calculate the sum

Calculate the sum of the terms: \(\frac{4}{3}+\frac{6}{3}+\frac{8}{3} = \frac{18}{3} = 6\) #f. Evaluating the expression#

Step 1: Write down the expression with each term

For f: \(\sum_{j=1}^{3}(3 j-4)\) , replace j with 1 to 3 and write down each term: \((-1)+2+5\)

Step 2: Calculate the sum

Calculate the sum of the terms: \((-1)+2+5 = 6\) #g. Evaluating the expression#

Step 1: Write down the expression with each term

For g: \(\sum_{p=1}^{5}\left(2 p+p^{2}\right)\) , replace p with 1 to 5 and write down each term: \(3+8+\cdots+61\)

Step 2: Calculate the sum

Calculate the sum of the terms: \(3+8+\cdots+61 = 115\) #h. Evaluating the expression#

Step 1: Write down the expression with each term

For h: \(\sum_{n=0}^{4} \sin \frac{n \pi}{2}\) , replace n with 0 to 4 and write down each term: \(\sin(0)+\sin\left(\frac{\pi}{2}\right)+\sin(\pi)+\sin\left(\frac{3\pi}{2}\right)+\sin(2\pi)\)

Step 2: Calculate the sum

Calculate the sum of the terms: \(\sin(0)+\sin\left(\frac{\pi}{2}\right)+\sin(\pi)+\sin\left(\frac{3\pi}{2}\right)+\sin(2\pi) = 0+1+0-1+0 = 0\)

Key Concepts

Summation Formula

The summation formula provides a concise way to express the addition of a series of numbers. It is denoted by the Greek letter sigma, \( \Sigma \), which represents the process of adding up a sequence of terms. For example, the expression \( \sum_{k=1}^{10} k \) means that we are summing all the integers from 1 through 10. To calculate this, one could perform the addition manually as \(1+2+3+\cdots+10\), or use a summation formula for arithmetic series \(\frac{n(n+1)}{2}\), where \(n\) is the last number in the range. In this case, inputting \(10\) into the formula gives us \(\frac{10(10+1)}{2}=55\), which is the sum of the first ten positive integers.

Sequence of Numbers

A sequence of numbers is a specific order of numbers that follow a certain rule, which allows each member of the sequence to be predictably determined from one or more preceding numbers. Sequences can be finite, like the numbers from 1 to 10, or infinite, such as the sequence of all positive integers. The sequence can follow a simple rule like 'add 1 to the previous number' or more complex ones involving other arithmetic or algebraic operations. In the context of sigma notation, we often deal with sequences that form the terms of a series to be summed.

Arithmetic Sequences

An arithmetic sequence is a series of numbers in which each term after the first is created by adding a constant, known as the common difference, to the previous term. The sigma notation \(\sum_{k=1}^{6}(2k+1)\) is an example of an arithmetic series, where the sequence of numbers \(3, 5, 7, 9, 11, 13\) is formed by starting at 3 and repeatedly adding 2. To quickly calculate the sum of an arithmetic series, one can use the formula \(\frac{n}{2}(a_1 + a_n)\), where \(n\) is the number of terms, \(a_1\) the first term, and \(a_n\) the last term.

Geometric Sequences

In contrast, a geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. An example would be the sequence \(2, 4, 8, 16, \ldots\), where each number is twice the previous one, giving a common ratio of 2. Geometric sequences can be summed using sigma notation and the formula for the sum of a geometric series, \(S_n = a_1(1-r^n)/(1-r)\), where \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, and \(r\) is the common ratio.

Series Calculation

Series calculation refers to the process of finding the sum of the terms of a sequence. With sigma notation, it streamlines this task by indicating the start and end points of the sequence and the rule for creating each term. For example, the summation \(\sum_{k=1}^{4} k^2\) involves the sum of squares of the first four integers. To calculate this by hand, you'd add \(1^2 + 2^2 + 3^2 + 4^2\), which is 30. However, for longer sequences, using methods such as the arithmetic or geometric series formulas or the properties of sigma notation can greatly simplify the process.