Problem 19 - Calculus: Early Transcendental Functions

Question

Compute the sum and the limit of the sum as \(n \rightarrow \infty.\) $$\sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{i}{n}\right)^{2}+2\left(\frac{i}{n}\right)\right]$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{(n+1)(2n+1)}{6n^2} + \frac{(n+1)}{n} \), and the limit of this sum, as n approaches infinity, is \( \frac{4}{3} \).

Step 1: Analyzing the Sum

Break down the summation into two parts by separating the two terms in the brackets, both multiplied by \( \frac{1}{n} \). That results in: \( \frac{1}{n} \sum_{i=1}^{n} \left( \frac{i^{2}}{n^{2}} \right) + \frac{1}{n} \sum_{i=1}^{n} \left( \frac{2i}{n} \right) \).

Step 2: Evaluate the Sums

Now, calculate each sum separately. For the first sum, we use the formula for the sum of squares of the first n natural numbers, \( \frac{n(n+1)(2n+1)}{6} \), and simplify the sum to get \( \frac{1}{n^3} * \frac{n(n+1)(2n+1)}{6} = \frac{(n+1)(2n+1)}{6n^2} \). For the second sum, we use the formula for the sum of first n natural numbers, \( \frac{n(n+1)}{2} \), and simplify the sum to get \( \frac{2}{n^2} * \frac{n(n+1)}{2} = \frac{(n+1)}{n} \). So, the total sum becomes \( \frac{(n+1)(2n+1)}{6n^2} + \frac{(n+1)}{n} \).

Step 3: Compute the Limit

Use properties of limits to compute the limit as n approaches infinity. Treat each term of the sum separately. The limit as n approaches infinity is given by the ratio of leading coefficients for each term when n gets very large. Therefore, as n approaches infinity, the limit of the sum \( \frac{(n+1)(2n+1)}{6n^2} + \frac{(n+1)}{n} \) is \( \frac{2}{6} + 1 = \frac{4}{3} \).

Key Concepts

Infinite Series

An infinite series involves the sum of infinitely many numbers, typically expressed as a sequence of terms. Calculating the sum of an infinite series is not as straightforward as summing a finite list of numbers. To determine whether an infinite series has a well-defined sum, that is, if it converges, mathematicians rely on different tests and properties of limits.

• For example, a common method is to look at the limit of the partial sums of the series as the number of terms increases indefinitely. If this limit exists and is finite, the series is said to converge to that value.
• Another concept is the geometric series, which has a fixed ratio between successive terms, and for which a formula exists to find the sum if the series is convergent.

Understanding infinite series is crucial for domains like calculus, physics, and engineering, where they are commonly used to represent functions or solve complex problems.

Sum of Natural Numbers

The sum of natural numbers involves adding a sequence of numbers starting from 1 incrementally by 1, such as 1, 2, 3, and so forth, up to a certain number, n. Formulas exist for calculating these sums directly without having to add each individual number.

Common Formulas

• The sum of the first n natural numbers is given by the formula: \[ \frac{n(n+1)}{2} \]
• Similarly, the sum of the squares of the first n natural numbers is calculated using: \[ \frac{n(n+1)(2n+1)}{6} \]

These formulas are derived from mathematical inductions and are essential in simplifying complex problems in calculus - such as the one in our exercise - where these sums appear in the analysis of series or integrals.

Properties of Limits

The properties of limits are rules that make finding the limits of functions easier, especially when dealing with complex expressions. Limits describe the behavior of a function as the input approaches a certain value, which in the case of sum computations, is typically infinity.

• A key rule is the limit of a constant times a function is the constant times the limit of the function, provided the limit exists.
• The limit of a sum is the sum of the limits, again if the limits exist.
• For rational functions, as the variable approaches infinity, the limit is determined by the ratio of the leading coefficients of the highest powers of the variable.

These properties are elegantly showcased in the exercise example, which uses them to compute the limit of each sum separately as n approaches infinity.