MATH1051 Lecture Notes - Lecture 23: Alternating Series, Ratio Test, Nth Root
Power Series
What is a Power series?
●They’re functions defined as series
●(infinite degree “polynomial”)(x) xf = ∑
∞
n=0
ann
●What do we want from power series?
○What is the domain of the power series?
○In other words, what numbers would cause the series to converge
○What is the interval of convergence
Example #1
●∑
∞
n=0
xn
○We don’t always know what x is
■So the first thing to do is ALWAYS the Ratio or Root test
○Root test
■ lim
n→∞ √
na
|n|
■ lim
n→∞ √
nx
|n|
■lim
n→∞ x
| |
■Rx
| | =
○It will converge if x
| | < 1
■When − 1 < x< 1
○Diverge if x
| | > 1
■When or x−x> 1 < 1
○And we can’t draw a conclusion if x
| | = 1
■If 1 or x= − 1
○What happens at the points themselves though?
■At 1x=
●(1)∑
∞
n=0
n
○Plug in the point for x
●(1)∑
∞
n=0
○This is divergent, so we do NOT include the point in the
interval of convergence
■At x = -1
●(− )∑
∞
n=0
1n
○Alternating series
●fails the first test 0lim
n→∞ 1 =
●? yesan+1 ≤an
●However, for the series to pass the alternating series test, it must
pass both conditions
●So we don’t count this as a pass
■And therefore, don’t count this point in the interval of convergence
●oC (− , 1)I: 1
Example #2
● = Root test∑
∞
n=1 n2
(x−1)n
○ lim
n→∞ √
n
n2
(x−1)n
○lim
n→∞ √
nn2
√
n(x−1)n
○lim
n→∞ √
nn2
x−1
| |
○Rx
|− 1| =
■0 < x< 2
Document Summary
They"re functions defined as series f (x) n=0 an n x (infinite degree polynomial ) In other words, what numbers would cause the series to converge. We don"t always know what x is. So the first thing to do is always the ratio or root test. Root test n n a| n| lim n n x| n| lim. When x > 1 or x . And we can"t draw a conclusion if x| What happens at the points themselves though? n. Plug in the point for x (1) n=0. This is divergent, so we do not include the point in the interval of convergence. However, for the series to pass the alternating series test, it must pass both conditions. So we don"t count this as a pass. And therefore, don"t count this point in the interval of convergence. N (x 1)n n n n2 x 1 n n n2 x| 1| = Meaning we include 2 in our interval.
