Mastering Series and Sequences

Series and Sequences

• This chapter focuses almost exclusively on series but also covers some basics of sequences.
• Series play an important role in ordinary and partial differential equations.
• The chapter covers sequences, limits, convergence, monotonicity, and boundedness.
• It also covers the basics of infinite series and ways to manipulate them.
• The chapter discusses convergence and divergence of infinite series using partial sums and the Divergence Test.
• Three special series, Geometric, Telescoping, and Harmonic, are examined in detail.
• The Integral Test, Comparison Test, Limit Comparison Test, Alternating Series Test, Ratio Test, and Root Test are discussed with proofs.
• The chapter provides a general strategy for determining which test to use and a summary of all the tests.
• The chapter covers estimating the value of a series and power series with definitions of the radius and interval of convergence.
• The chapter discusses how the formula for a convergent Geometric Series can be used to represent some functions as power series and differentiation and integration of power series.
• The chapter covers finding the Taylor/Maclaurin Series for a function and deriving formulas for ({\bf e}^{x}) , (\cos(x)) and (\sin(x)) around (x=0).
• The chapter concludes with a quick look at applications of series, including finding a series representation for indefinite integrals and approximating a function using the first few terms of a power series.

Test your knowledge on series and sequences with this quiz! From understanding the basics of convergence, monotonicity, and boundedness to manipulating infinite series, this quiz covers it all. You'll also have the chance to examine special series, proofs for different tests, and strategies for determining which test to use. Additionally, the quiz includes questions on estimating the value of a series, power series, and finding Taylor/Maclaurin series for a function. Put your skills to the test and see how well