Problem 36 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The formula for the sum of \(n\) terms is \(2\frac{n(n+1)}{n^2}\), and the limit as \(n \rightarrow \infty\) is equal to 2.

Step 1: Finding the formula for the sum

First, write out the formula for the sum of \(n\) terms of the given sequence. In this case, the general term of the sequence can be written as \(\frac{2i}{n}\cdot\frac{2}{n} = \frac{4i}{n^2}\). The sum of \(n\) terms can be written as \(\sum_{i=1}^{n}\frac{4i}{n^2} = \frac{4}{n^2}\sum_{i=1}^{n}i\).

Step 2: Substitute the formula for the sum of first \(n\) natural numbers

Using the formula for the sum of the first \(n\) natural numbers, which is \(\frac{n(n+1)}{2}\), the sum can be written as \(\frac{4}{n^2}\cdot\frac{n(n+1)}{2} = 2\frac{n(n+1)}{n^2}\). Simplifying this further gets \(\frac{2}{n}\) for large \(n\). This is the sum formula needed for the limit.

Step 3: Find the limit as \(n \rightarrow \infty\)

With the formula for the sum of terms, apply the limit as \(n\) approaches infinity. This calls for the calculation of \(\lim_{n \rightarrow \infty} 2\frac{n(n+1)}{n^2}\). When taking the limit, the highest power in the denominator and numerator will dominate, so it simplifies to \(\lim_{n \rightarrow \infty} 2\frac{n^2}{n^2} = 2\).

Key Concepts

Convergence of Series

Understanding the convergence of series is like solving a puzzle where the pieces are numbers getting closer to a specific value. Simply put, when we talk about a series converging, we mean that as we keep adding terms, the total sum approaches a particular number, known as the limit. This is a crucial concept in calculus and analysis because not all series will neatly sum up to a precise number. Some series keep growing indefinitely, and those are said to diverge.

Imagine you're saving money and with each month, you save half of what you saved in the previous month. Initially, it might look like you'll accumulate infinite funds over time, but in reality, you're getting closer to a finite amount. This scenario is similar to a convergent series where each term adds a smaller and smaller amount to the total sum. When the exercise asks for the limit as \(n \rightarrow \infty\), it challenges us to determine this finite amount that the series is approaching - the point where adding more terms doesn't significantly change the sum.

Riemann Sum

Imagine slicing an apple into thinner and thinner pieces, trying to determine the total volume of the apple. The Riemann sum is akin to adding up the volumes of these slices to approximate the volume of the apple. It’s a method used in calculus to estimate the total area under a curve. To form a Riemann sum, we divide the area into rectangles (or occasionally other shapes), find the area of each one, and then add them all up. As the width of these rectangles becomes increasingly small, the sum becomes a better and better approximation of the actual area.

In our exercise, each term of the sum \(\left(\frac{2i}{n}\right)\left(\frac{2}{n}\right)\) is like the area of one of these rectangles. As \(n\) gets very large, the width of each rectangle \(\left(\frac{2}{n}\right)\) gets very small, and the Riemann sum approximates the area under the curve of the function \(f(i) = \frac{2i}{n}\) on the interval from 1 to \(n\). This technique isn't just for areas—it's a foundational tool for integrating functions over an interval.

Infinite Series

An infinite series is like a never-ending domino chain, where each domino represents a term in the series. If we were to tip the first domino, the chain would continue falling indefinitely. Likewise, an infinite series is an expression that keeps on adding terms, theoretically forever. But the real question is: does knocking over these infinitely many dominos amount to anything finite, or does it go on without bound?

In mathematics, determining the sum of an infinite series is a delicate task. Sometimes, adding up an infinite number of terms results in a finite number, just like how endless falling dominos might travel a finite distance if each domino is closer to the next than the one before it. This finite sum is what we seek in many calculus problems. For example, the series in our exercise can be used to compute such a sum, as long as it converges. The limit of the sum formula allows us to peek at the endpoint of this infinite process: \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(\frac{2i}{n}\right)\left(\frac{2}{n}\right)\) tells us the finite destination where the infinite number of terms collectively leads us.