Basic Calculus Review: Infinite Series

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Basic Calculus Review: Infinite Series

Presentation transcript:

Basic Calculus Review: Infinite Series Suppose an ant starts at the left end of a 12-inch ruler, and walks to the right end.

Basic Calculus Review: Infinite Series Suppose an ant starts at the left end of a 12-inch ruler, and walks to the right end. She then about-faces, and walks half the previous distance…

Basic Calculus Review: Infinite Series Suppose an ant starts at the left end of a 12-inch ruler, and walks to the right end. She then about-faces, and walks half the previous distance… and continues ad infinitum, each time reversing direction, and walking half the previous distance.

Basic Calculus Review: Infinite Series Suppose an ant starts at the left end of a 12-inch ruler, and walks to the right end. She then about-faces, and walks half the previous distance… and continues ad infinitum, each time reversing direction, and walking half the previous distance.

Basic Calculus Review: Infinite Series Suppose an ant starts at the left end of a 12-inch ruler, and walks to the right end. She then about-faces, and walks half the previous distance… and continues ad infinitum, each time reversing direction, and walking half the previous distance. etc. Questions: Both of these are examples of geometric series. What is the total distance the ant walks? Formal generalization? Where on the ruler does she “settle” eventually?

Basic Calculus Review: Infinite Series Def: An infinite series of is the summation Note: We are usually only interested in infinite series that converge (to a unique, finite value). A finite series has the form (n + 1 terms) We may thus define the infinite series as provided that the limit exists! Example: Let a and r be any real constants. Def: A finite geometric series has the form r is called the common ratio Can this sum be expressed in explicit, closed form? ↑ first term

Basic Calculus Review: Infinite Series Example: Let a and r be any real constants. Def: A finite geometric series has the form Exercise: What happens if r = 1? What about the infinite geometric series?

Basic Calculus Review: Infinite Series Suppose an ant starts at the left end of a 12-inch ruler, and walks to the right end. She then about-faces, and walks half the previous distance… and continues ad infinitum, each time reversing direction, and walking half the previous distance. etc. Questions: Both of these are examples of geometric series. What is the total distance the ant walks? Where on the ruler does she “settle” eventually?

Basic Calculus Review: Infinite Series Example of a polynomial of degree n Example of a power series

Basic Calculus Review: Infinite Series

Basic Calculus Review: Infinite Series

Basic Calculus Review: Infinite Series

Basic Calculus Review: Infinite Series Taylor polynomials Taylor series expansion for around x = 0 etc.

Basic Calculus Review: Infinite Series Taylor polynomials Taylor series expansion for around x = 0 Taylor series expansion for around x = 0

Basic Calculus Review: Infinite Series Taylor polynomials Taylor series expansion for around x = 0 Taylor series expansion for around x = 0 Taylor series expansion for around x = 0

Basic Calculus Review: Infinite Series Taylor polynomials Taylor series expansion for around x = 0 Taylor series expansion for around x = 0 Taylor series expansion for around x = 0

Basic Calculus Review: Infinite Series Find the power series expansions of the left- and right-hand sides separately, and show agreement. Exercises: Find the power series expansion of. Find the power series expansion of.

BINOMIAL THEOREM Basic Calculus Review: Infinite Series Taylor series for f(x) around x = 0 (a.k.a. Maclaurin series for f(x)) Recall that for any positive integer n, “n factorial” = n! = 1  2  3  …  n. Others… “binomial coefficients” “combinatorial symbols” Read this review document. In general… BINOMIAL THEOREM