Problem 31 - Calculus: One and Several Variables

Question

Let \(\sum_{i=0}^{\infty} a_{z}\) be a convergent series and let \(R_{o}=\sum_{j=n+1}^{\infty} a_{k} .\) Prove that \(R_{n} \rightarrow 0\) as \(n \rightarrow \infty\). Note that if \(s_{n}\) is the nth partial sum of the series, then \(\sum_{k=9}^{\infty} a_{k}=s_{n}+R_{m} ; R_{n}\) is called the remainder.

Step-by-Step Solution

Answer

Since the series is convergent, it has a finite limit L: \(\lim_{n\rightarrow\infty} s_n = L\). We can rewrite the remainder as the difference between the total sum and the nth partial sum: \(R_n = L - s_n\). To prove that the remainder approaches 0, we need to determine the limit of \(R_n\) as n approaches infinity: \(\lim_{n\rightarrow\infty} R_n = \lim_{n\rightarrow\infty} (L - s_n) = L - L = 0\). Therefore, the remainder of a convergent series approaches 0 as the number of terms increases.

Step 1: Define the nth partial sum and the total sum

Given a convergent series \(\sum_{i=0}^{\infty} a_{i}\), let the nth partial sum be: \[s_{n} = \sum_{i=0}^{n} a_{i}\] As the series is convergent, it has a finite limit L: \[\lim_{n\rightarrow\infty} s_n = L\] We need to prove that the remainder approaches 0, which can be represented as: \[R_n = \sum_{i=n+1}^{\infty} a_i\]

Step 2: Express remainder in terms of partial sum

We can rewrite the remainder as the difference between the total sum and the nth partial sum: \[R_n = L - s_n\] Our goal is to show that \(R_n \rightarrow 0\) as \(n \rightarrow \infty\).

Step 3: Calculate the limit of the remainder

To prove that the remainder approaches 0, we need to determine the limit of \(R_n\) as n approaches infinity: \[\lim_{n\rightarrow\infty} R_n = \lim_{n\rightarrow\infty} (L - s_n)\] Since we know the limit of s_n: \[\lim_{n\rightarrow\infty} s_n = L\] We can use the limit laws to evaluate the limit of the difference: \[\lim_{n\rightarrow\infty} R_n = L - \lim_{n\rightarrow\infty} s_n\] \[\lim_{n\rightarrow\infty} R_n = L - L\] \[\lim_{n\rightarrow\infty} R_n = 0\]

Step 4: Conclusion

We have shown that the limit of the remainder, \(R_n\), approaches 0 as n approaches infinity. Therefore, the remainder of a convergent series approaches 0 as the number of terms increases.

Key Concepts

Partial Sum of a Series

The partial sum of a series is a fundamental concept when dealing with infinite series. A partial sum, denoted as \(s_n\), is simply the sum of the first \(n\) terms of an infinite series. Thus, if we have a series \(\textstyle\frac{1}{1} \frac{1}{2}\), \(\textstyle\frac{1}{4}\), ... , the \(n\)th partial sum \(s_n\) would be the sum of the first \(n\) fractions.

To illustrate, consider the series \(\textstyle\frac{1}{1} \frac{1}{2}\) \(\textstyle\frac{1}{4}\), ..., the first partial sum \(s_1\) would be \(1\), the second partial sum \(s_2\) would be \(1 \frac{1}{2}\), and so on. The study of partial sums allows us to understand the behavior of infinite series by examining the properties and trends of finite sections within them.

In relation to convergence, if a series converges, its partial sums will approach a specific finite value, known as the limit of the series, as \(n\) becomes larger. This is why exploring partial sums is crucial in determining convergence or divergence of an infinite series.

Limit of a Sequence

The concept of the limit of a sequence is essential in understanding mathematical convergence. A sequence \(a_n\) has a limit \(L\) if, as \(n\) increases without bound, the terms of the sequence get arbitrarily close to \(L\). In formal mathematical terms, \(\textstyle\frac{1}{n} \frac{1}{L}\) given any positive number \(\textstyle\frac{1}{\varepsilon}\), there exists a natural number \(\textstyle\frac{1}{N}\) such that for all \(n \geq \textstyle\frac{1}{N}\), the distance between \(a_n\) and \(L\) is less than \(\textstyle\frac{1}{\varepsilon}\).

In the context of series, if we have a convergent series, then the limits of its partial sums (the sequence of partial sums) will converge to a specific value. This is what we explore when we say a series \(\textstyle\frac{1}{n} \frac{1}{L}\) converges. In simpler terms, as you add more and more terms of the series, the sum will stabilize at a particular value, which is the limit of the sequence of partial sums.

Remainder of a Series

The remainder of a series often becomes a point of interest when evaluating the convergence of an infinite series. It is defined as the difference between the total sum of an infinite series, if it exists, and the sum of the first \(n\) terms. In symbols, if \(R_n\) denotes the remainder after the \(n\)th term, it can be expressed as the difference between the total sum \(L\) and the partial sum \(s_n\): \(R_n = L - s_n\).

The concept of the remainder is important because it effectively measures how much the sum of the first \(n\) terms is 'off' from the total sum. In a convergent series, as \(n\) tends to infinity, we expect the remainder to trend towards zero. This implies that adding more terms to the partial sum does not significantly change the sum, reinforcing the idea that the partial sum is an good approximation of the total sum. In our exercise, proving \(R_n \rightarrow \textstyle\frac{1}{0}\) as \(n \rightarrow \textstyle\frac{1}{\infty}\) confirms that the series has a finite limit. This notion of the remainder is invaluable in practical situations, such as numerical analysis and approximations of complex functions.