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

2. 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…

3. 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.

4. 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.

5. 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?

6. 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

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

8. 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?

9. Basic Calculus Review: Infinite Series
Example of a polynomial of degree n
(A finite sum of non-negative powers of x.)
Example of a power series
(An infinite sum of non-negative powers of x.)
NOT a polynomial!

10. Basic Calculus Review: Infinite Series

11. Basic Calculus Review: Infinite Series

12. Basic Calculus Review: Infinite Series

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

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

15. 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

16. 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

17. 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.

18. 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