Question

Suppose that $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ is a convergent series of positive terms. Explain why $\underset{N\to \mathrm{\infty }}{lim}\sum _{n=N+1}^{\mathrm{\infty }}{a}_n=0$.

Short answer

The remainder of a convergent series, after any finite number of terms, approaches zero as the number of terms goes to infinity.

Step-by-step solution

Understanding Convergent Series

A series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ is said to be convergent if its sequence of partial sums $S_N=\sum _{n=1}^N{a}_n$ approaches a finite limit as $N$ approaches infinity. This means that there exists a sum $S$ such that $\underset{N\to \mathrm{\infty }}{lim}{S}_N=S$.

Remainder of the Series

In order to understand why $\underset{N\to \mathrm{\infty }}{lim}\sum _{n=N+1}^{\mathrm{\infty }}{a}_n=0$, we consider this sum as the remainder of the series after finitely many terms. It represents $S-S_N$, the difference between the total sum of the series and the partial sum up to $N$.

Limit of the Remainder

The series converges, so $\underset{N\to \mathrm{\infty }}{lim}{S}_N=S$. Therefore, as $N$ increases, $S_N$ gets closer and closer to $S$. The difference between the complete sum $S$ and $S_N$ should decrease, meaning $\sum _{n=N+1}^{\mathrm{\infty }}{a}_n=S-S_N\to 0$ as $N\to \mathrm{\infty }$.

Conclusion

Since the series is convergent, the remainder of the series past $N$ terms approaches zero, demonstrating that the series sums to a finite value. Hence, $\underset{N\to \mathrm{\infty }}{lim}\sum _{n=N+1}^{\mathrm{\infty }}{a}_n=0$.

Key concepts

Partial Sums

In the context of a convergent series, a partial sum is the sum of the first few terms. If we have a series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$, the partial sum $S_N$ is the sum of the first $N$ terms, such that:

• $S_N=\sum _{n=1}^N{a}_n$

This concept is fundamental because it allows us to understand how the series behaves as we add more and more terms. By looking at the sequence of partial sums, we can determine if the series converges or diverges.
For a series to be convergent, its partial sums must approach a specific number, also known as the limit of the series. This means $S_N$ gets closer and closer to some finite sum $S$ as $N$ becomes very large. This pattern in the sequence of partial sums is what helps us establish the convergence and the ultimate value that the series approaches.

Limit of a Series

The limit of a series is essentially the value that the sequence of its partial sums approaches as the number of terms goes to infinity. Let’s denote this finite value as $S$. For a given convergent series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$, we say that its limit is $S$ if:

• $\underset{N\to \mathrm{\infty }}{lim}{S}_N=S$

This means that no matter how far we go to add more terms past a certain point, the changes in the sum become negligible and do not affect the result significantly.
This concept is crucial because it defines the behavior of the series at infinity. Essentially, it assures us that the infinite process of adding terms does not lead to an undefined or infinite result, but to a stable and meaningful sum. Understanding the limit of a series provides assurance about the nature and the endpoint of what would otherwise seem like an endless progression.

Remainder of a Series

Understanding the remainder of a series helps us see what happens to the terms after a certain point. In a series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$, the remainder after adding the first $N$ terms can be represented as $\sum _{n=N+1}^{\mathrm{\infty }}{a}_n$. This sum is the remainder and equals:

• $R_N=S-S_N$

where $S$ is the limit of the series, and $S_N$ is the $N$-th partial sum.
In a convergent series, this remainder $R_N$ becomes very small as $N$ increases, approaching zero. Mathematically, we express this as:

• $\underset{N\to \mathrm{\infty }}{lim}{R}_N=0$

This indicates that most of the series' total sum is captured in its early terms, confirming that later terms add negligible amounts, allowing the series to converge. It reassures us that any further terms past a point barely affect the overall value of the series, reinforcing why convergent series reach a stable and finite total.