Problem 3 - Calculus: Early Transcendental Functions

Question

A calculation is described in words. Translate each into summation notation and then compute the sum. The sum of the square roots of the first 10 positive integers.

Step-by-Step Solution

Answer

The sum of the square roots of the first 10 positive integers is approximately 22.4682781862.

Step 1: Translation to Summation Notation

The statement describes the sum of the square roots of the first 10 positive integers. In summation notation, this can be represented as \( \sum_{i=1}^{10} \sqrt{i} \), where i is the index of summation that ranges from 1 to 10.

Step 2: Computation of the Sum

Next, compute the sum \( \sum_{i=1}^{10} \sqrt{i} = \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} \). Use a calculator or mathematical resources to find the square root of each integer and then add these values together.

Step 3: Final Sum

After completing the computation, you should find that the sum equals \(\sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} = 22.4682781862.\) Note that the result is rounded to 10 decimal places for accuracy.

Key Concepts

Square Roots

Square roots are mathematical operations that answer the question: what number, when multiplied by itself, will give the original number? For instance, the square root of 9 is 3, because 3 times 3 equals 9. In mathematical notation, this is expressed as \( \sqrt{9} = 3 \).

This concept is fundamental in many areas of mathematics including algebra and geometry. In the context of our exercise, understanding square roots is crucial because it involves the operation of taking square roots of integers. It's also important to note that each positive integer has a unique non-negative square root. When solving problems involving square roots, you may sometimes need a calculator, especially for non-perfect squares like \( \sqrt{2} \).

Positive Integers

Positive integers are all the whole numbers greater than zero (1, 2, 3, ...). They are the most basic numerical building blocks and are used to count, order, and label. In our exercise, we focus on the first ten positive integers. They form a sequence, which is a list of numbers in a specific order. Sequences play a core role in fields like number theory and combinatorics. When dealing with summation problems, it's important to identify the sequence of terms involved and their range — in this case, from 1 to 10.

Calculus

Understanding Summation Notation through Calculus

Calculus is a branch of mathematics centered around the study of change (differential calculus) and accumulation (integral calculus). Integral calculus, in particular, uses a form of summation notation to represent areas and accumulated quantities. While the concept of summation in calculus might deal with an infinite number of terms, our exercise involves a finite series. Nonetheless, familiarity with calculus helps to understand the representation and manipulation of summations like the one in our exercise, \( \sum_{i=1}^{10} \sqrt{i} \).

Series and Sequences

A sequence is an ordered list of numbers, and a series is the sum of a sequence of numbers. In simple terms, if a sequence is the list of ingredients, then a series is the final dish. In our exercise, we deal with a very specific type of series where each term is the square root of a positive integer. When working with series, it's important to understand the relationship between the terms and how to express the series using summation notation, as it provides a concise and clear way to communicate what we are summing. Summation notation also makes it easier to apply formulas or computational techniques to find the sum.