Convergent Series

Definition

A convergent series is a type of infinite series where the sum of the terms approaches a finite limit as the number of terms increases. This means that the series has a well-defined sum, and the partial sums of the series converge to this limit.

5 Must Know Facts For Your Next Test

1. The limit of a convergent series is called the sum of the series, and it represents the value that the partial sums approach as the number of terms increases.
2. The Divergence Test and the Comparison Test are two important tests used to determine whether a series is convergent or divergent.
3. Absolutely convergent series are a subset of convergent series, and they have the property that the series will converge regardless of the order in which the terms are added.
4. Convergent series can be used to approximate the values of certain functions, such as $e^x$ and $\sin(x)$, by representing them as infinite series.
5. The Alternating Series Test can be used to determine the convergence of alternating series, which are a type of convergent series where the terms alternate in sign.

Review Questions

• Explain the difference between a convergent series and a divergent series, and provide an example of each.
  • A convergent series is an infinite series where the sum of the terms approaches a finite limit as the number of terms increases. This means that the partial sums of the series converge to a well-defined value. In contrast, a divergent series is an infinite series where the sum of the terms does not approach a finite limit, and the partial sums continue to grow without bound. An example of a convergent series is the geometric series $\sum_{n=0}^{\infty} \frac{1}{2^n}$, which converges to 2. An example of a divergent series is the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$, which diverges.
• Describe the concept of absolute convergence and its importance in the study of convergent series.
  • Absolute convergence is a stronger form of convergence for infinite series. A series is said to be absolutely convergent if the series formed by the absolute values of its terms is convergent. Absolutely convergent series have the property that they will converge regardless of the order in which the terms are added. This is important because it ensures that the sum of the series is well-defined and does not depend on the way the terms are grouped or rearranged. Absolute convergence is a desirable property for series, as it allows for more flexibility in working with the series and ensures that the series behaves in a more predictable manner.
• Explain how convergent series can be used to approximate the values of certain functions, and discuss the significance of this application.
  • Convergent series can be used to represent and approximate the values of certain functions, such as $e^x$ and $\sin(x)$, by expressing them as infinite series. This is significant because it allows us to calculate the values of these functions to any desired degree of accuracy by considering the partial sums of the series. The more terms we include in the partial sum, the closer the approximation will be to the true value of the function. This is particularly useful in areas like numerical analysis and scientific computing, where we often need to evaluate functions quickly and efficiently. By using convergent series, we can obtain accurate approximations without having to resort to more complex or computationally intensive methods.

Related terms

Divergent Series: A divergent series is an infinite series where the sum of the terms does not approach a finite limit as the number of terms increases. The partial sums of a divergent series continue to grow without bound.

Partial Sum: The partial sum of a series is the sum of the first $n$ terms of the series. As $n$ increases, the partial sums of a convergent series approach the limit of the series.

Absolute Convergence: A series is said to be absolutely convergent if the series formed by the absolute values of its terms is convergent. Absolutely convergent series have stronger convergence properties than conditionally convergent series.