TEAM PROJECT. Properties from Power Series. In the next sections we shall define new

Chapter 5, Problem 5.1.42

(choose chapter or problem)

Properties from Power Series. In the next sections we shall define new functions (Legendre functions, etc.) by power series. deriving properties of the functions directly from the series. To understand this idea, do the same for functions familiar from calculus, using Maclaurin series.

(a) Show that \(\cosh x+\sinh x=e^{x}\). Show that cosh x > 0 for all x. Show that \(e^{x} \geqq e^{-x}\) for all \(x \geqq 0\).

(b) Derive the differentiation formulas for \(e^{x}\), cos x, sin x .1 /(1-x) and other functions of your choice. Show that (cos x) “ = -cos x, (cosh x) “ = cosh x. Consider integration similarly.

(c) What can you conclude if a series contains only odd powers? Only even powers? No constant term? If all its coefficients are positive? Give examples.

(d) What properties of cos x and sin x are not obvious from the Maclaurin series? What properties of other functions?

Text Transcription:

cosh x + sinh x = e^x

e^x geqq e^-x

x geqq 0

e^x