Lesson 482 of 1524
Power Series
For a power series centered at x = a , one of the following three properties hold: The power series converges only at x = a .
Practice this chapterFor a power series centered at x = a , one of the following three properties hold: The power series converges only at x = a . In this case, we say that the radius of convergence is R = 0
The power series converges for all real numbers x . In this case, we say that the radius of convergence is R = ∞
power series — a series of the form ∑ n = 0 ∞ c n x n is a power series centered at x = 0 ; a series of the form ∑ n = 0 ∞ c n ( x - a ) n is a power series centered at x = a. radius of convergence — if there exists a real number R > 0 such that a power series centered at x = a converges for | x - a | < R and diverges for | x - a | > R , then R is the radius of convergence…. binomial series — the Maclaurin series for f ( x ) = ( 1 + x ) r ; it is given by ( 1 + x ) r = ∑ n = 0 ∞ ( r n ) x n = 1 + r x + r ( r - 1 ) 2 ! x 2 + ⋯ + r ( r - 1 ) ⋯ ( r - n + 1 ) n ! x n + ⋯…. Taylor polynomials — the n th Taylor polynomial for f at x = a is p n ( x ) = f ( a ) + f ′ ( a ) ( x - a ) + f ″ ( a ) 2 ! ( x - a ) 2 + ⋯ + f ( n ) ( a ) n ! ( x - a ) n p n ( x ) = f ( a ) + f ′ (….
Worked example
What does “power series” mean in Power Series?
- 1Use the wording this chapter gives for power series.
- 2The book says: a series of the form ∑ n = 0 ∞ c n x n is a power series centered at x = 0 ; a series of the form ∑ n = 0 ∞ c n ( x - a ) n is a power series centered at x = a.
- 3Do not use the meaning of radius of convergence. That term means if there exists a real number R > 0 such that a power series centered at x = a converges for | x - a | < R and diverges for | x - a | > R , then R is the radius of convergence….
Result: a series of the form ∑ n = 0 ∞ c n x n is a power series centered at x = 0 ; a series of the form ∑ n = 0 ∞ c n ( x - a ) n is a power series centered at x = a
Why. That is the meaning this chapter gives for power series.
Do not swap power series and radius of convergence. power series means a series of the form ∑ n = 0 ∞ c n x n is a power series centered at x = 0 ; a series of the form ∑ n = 0 ∞ c n ( x - a ) n is a power series centered at x = a. radius of convergence means if there exists a real number R > 0 such that a power series centered at x = a converges for | x - a | < R and diverges for | x - a | > R , then R is the radius of convergence….
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.