1. Power Series
Section 9.1a

2. Definition: Infinite Series
A finite sum of real numbers always produces a real number,
but an infinite sum of real numbers is not actually a real sum:
Definition: Infinite Series
An infinite series is an expression of the form or
The numbers are the terms of the series;
is the nth term.

3. A finite sum of real numbers always produces a real number,
but an infinite sum of real numbers is not actually a real sum:
The partial sums of the
series form a sequence
If the sequence of partial
sums has a limit S as n
approaches infinity, we say
the series converges to S:
Otherwise, we say the series diverges.

4. Guided Practice Does the given series diverge?
Can we use the associate property of addition?:
This does NOT work with infinite series!!!
Instead, look at the sequence of partial sums:
Since this sequence has no finite limit, then the original
series has no sum. It diverges.

5. Guided Practice Does the given series diverge?
Look at the sequence of partial sums:
This sequence has a limit , which we recognize as 1/3.
The series converges to the sum 1/3.
This last problem is an example of a geometric series,
because each term is obtained by multiplying the previous
term by the same constant (1/10 in this case)…

6. Geometric Series The geometric series converges to the sum if ,
and diverges if
The interval is the interval of convergence.
(these are the r values for which the series converges)

7. Guided Practice Tell whether each series converges or diverges. If it
converges, give its sum.
The first term is a = 3 and r = 1/2.
The series converges to:
The first term is a = 1
and r = –1/2.
The series converges to:

8. Guided Practice Tell whether each series converges or diverges. If it
converges, give its sum.
The first term is a = 9/25 and r = 3/5.
The series converges to:
In this series,
The series diverges.

9. Representing Functions by Series
If , then the geometric series formula gives:
The expression on the left
defines a function whose
domain is the interval of
convergence,
The expression on the right
defines a function whose
domain is the set of all real
numbers
The equality is understood to hold only in this latter domain,
where both sides of the equation are defined. On this
domain, the series represents the function 1/(1 – x).

10. Representing Functions by Series
If , then the geometric series formula gives:
Let’s graph the function together with some partial sums of
the series. Graph in [–4.7, 4.7] by [–2, 4]:
The graphs behave similarly, but only
on the interval of convergence!!!

11. Definition: Power Series
An expression of the form
is a power series centered at x = 0. An expression of the form
is a power series centered at x = a. The term is
the nth term; the number a is the center.

12. Guided Practice
Given that 1/(1 – x) is represented by the power series
on the interval (–1, 1), find a power series that represents 1. on 2. on 3. on

13. Guided Practice
Given that 1/(1 – x) is represented by the power series
on the interval (–1, 1), find a power series that represents 4.
and give its interval of convergence
This geometric series converges for:
Interval of convergence:

14. Guided Practice
Given that 1/(1 – x) is represented by the power series
on the interval (–1, 1), find a power series that represents 5.
and give its interval
of convergence
This geometric series
converges for:
Interval of
convergence: