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.