Question

In Exercises \(35-40,\) find a formula for the sum of \(n\) terms. Use the formula to find the limit as \(n \rightarrow \infty\). $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{i}{n}\right)\left(\frac{2}{n}\right) $$

Short answer

The limit as \(n\) tends to infinity of the given sum is 3.

Step-by-step solution

Simplify the Formula

The given mathematical expression consists of two multiplication terms: \(1+\frac{i}{n}\) and \(\frac{2}{n}\). Distributing \(\frac{2}{n}\) over \(1+\frac{i}{n}\), we get \(\frac{2}{n} + \frac{2i}{n^2}\). Thus, the mathematical expression becomes: \[\sum_{i=1}^{n}\left(\frac{2}{n} + \frac{2i}{n^2}\right)\]

Calculate the Sum of n Terms

Calculating the sum of the first term \(\frac{2}{n}\) from \(i=1\) to \(n\) yields \(\sum_{i=1}^{n}\frac{2}{n} = \frac{2n}{n} = 2\). For the second term, \(\frac{2}{n^2} \sum_{i=1}^{n} i\), we can use the formula for the sum of the first \(n\) natural numbers which is \(\frac{n(n+1)}{2}\).Substituting this into the sum, we get \(\frac{2}{n^2} \times \frac{n(n+1)}{2} = \frac{n+1}{n}\).The sum of \(n\) terms of the given mathematical expression therefore becomes:\[2 + \frac{n+1}{n}\]

Find the Limit as n tends to infinity

The limit of the sum of \(n\) terms as \(n\) tends to infinity is calculated as:\[\lim _{n \rightarrow \infty} (2 + \frac{n+1}{n} )\]As \(n\) becomes infinitely large, the term \(\frac{n+1}{n}\) approaches 1. Thus, the above limit becomes \(2 + 1 = 3\).

Key concepts

Infinite Series

An infinite series is, at its core, the sum of an infinite sequence of numbers. When we look at the series ewlineewlineewlineewlineewline \( \sum_{i=1}^{\infty} a_i \), we're considering the limit of the partial sums of sequence \( \{a_i\} \) as the number of terms goes to infinity.
In the context of our exercise, we seek to determine the limit as \( n \rightarrow \infty \) of a particular infinite series. To make sense of this series, we break it down term by term and analyze its behavior as \( n \) grows larger. Indeed, what might seem like a daunting task at first can often be simplified by looking for patterns or recognizing familiar forms, like geometric or arithmetic series.

Therefore, understanding an infinite series involves both comprehending how to sum a sequence and discerning the behavior of that sum as we consider more and more terms. This forms the basis of determining the convergence or divergence of the series, which tells us whether the sum approaches a finite limit or not.

Riemann Sums

Riemann sums are a bridge between finite sums and integrals. They help us approximate the area under a curve, which is essential for integration.
Let's consider our series ewlineewlineewlineewlineewline \( \sum_{i=1}^{n}\left(1+\frac{i}{n}\right)\left(\frac{2}{n}\right) \). If we plot each term \( \left(1+\frac{i}{n}\right)\left(\frac{2}{n}\right) \) as a rectangle's area over an interval, the height is given by \( 1+\frac{i}{n} \) and the width by \( \frac{2}{n} \). Together, these areas approximate the integral of the function \( 1+x \) from 0 to 2 as \( n \) approaches infinity. This is a classic use of Riemann sums turning discrete sums into continuous integrals.

The sum given is a specific example of a Riemann sum, which becomes a Riemann integral in the limit as \( n \rightarrow \infty \). The interplay between the finite and infinite, the discrete and the continuous, is a fundamental aspect of calculus and particularly critical in understanding Riemann sums.

Sequences and Series Calculus

Sequences and series are foundational to calculus. A sequence is simply an ordered list of numbers, and a series is the sum of a sequence's elements. While sequences are about the individual elements and their behaviors, series are interested in the aggregate — the sum.
In exercises like the one provided, the calculus of series often involves finding a general formula for the sum of the first \( n \) terms and then examining what happens as \( n \) becomes very large. This process typically includes utilizing known formulas for the sum of elements in a sequence, such as the sum of the first \( n \) natural numbers, \( \frac{n(n+1)}{2} \), to simplify the series before taking the limit.

Ultimately, sequences and series lead us to explore convergence and divergence, integral tests, comparison tests, and other methods for understanding the behavior of series. A deep grasp of these concepts is essential not just for solving textbook problems, but for making sense of phenomena expressed through series in both mathematics and applied sciences.