Description

One very useful application of differentiation, integration and improper integrals is the study of inifinte sequences and series. In this topic, we explore sequences and monotonic sequences. Then we explore infinite series and develop methods to determine if series are convergent or divergent using the Integral, Comparison, Ratio and Root Tests. We extend our investigation by studying Alternating Series and Absolute Convergence. We develop Power Series and explore differentiation and integration of power series. This leads to Taylor Series and Binomial Series.

Sections of this topic include:

1. Sequences

   We begin this study by exploring what is meant by infinite sequences. We learn how to find terms of sequences; find patterns of sequences and determine if infinite sequences are convergent or divergent.

2. Monotonic Sequences

   To further explore infinite sequences, we explore the notion of monotonic sequences where each successive term in a sequences is continually increasing (or decreasing) and discuss when the monotonic infinite sequence converges or diverges.

3. Infinite Series

   In this section, we next investigate adding terms of infinite sequences and determine if that sum (called an infinite series) adds up to a number (convergent) or does not add up to a number (divergent). We begin to develop methods to determine whether an infinite series is convergent or divergent.

4. Integral and Comparison Tests (Coming Soon)

   Here, we introduce several tests to determine if an infinite series is convergent or divergent. The integral test determines if the series is convergent if it's corresponding real function is convergent. We also explore both the Basic Comparison and Limit Comparison Tests by comparing terms of the given series to known convergent or divergent series.

5. Ratio and Root Tests

   In this section, we explore several more tests for determining if infinite series are convergent or divergent. The ratio test compares two successive terms of an infinite series as a ratio to determine convergence while the root test is used to find the nth root of a term of an infinite series to determine convergence.

6. Alternating Series and Absolute Convergence (Coming Soon)

   The last type of infinite series we investigate are where each term alternates in sign which are called Alternating Series. We investigate whether an alternating series is conditionally convergent or absolutely convergent.

7. Power Series (Coming Soon)

   A direct application of infinite series is to create Power Series which are infinite series with a variable. Here we investigate radius of convergence and various power series representations of functions.

8. Differentiation and Integration of Power Series (Coming Soon)

   We find that with Power Series that we can differentiate and integrate each term of the infinite series. We explore the application of using this approach to finding power series for functions such as the natural logarithmic function.

9. Taylor Series (Coming Soon)

   In this section, we investigate how to find power series representations for transcendental functions using Taylor Series (named after the English mathematician Brook Taylor) and Maclaurin Series (named after the Scottish mathematician Colin Maclaurin).

10. Binomial Series (Coming Soon)

    In this last section, we apply Taylor Series and Binomial Series to extending the Binomial Theorem for rational or negative powers.