Calculus 2 Cheat Sheet Page 15
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16B Veitch
Calculus 2 Study Guide
13. Estimating a Series
(a) Remainder Estimate for Integral Test:
If
a
is convergent, f (n) is continuous,
n
n=1
positive, and decreasing, then
R
f (n) dn
N
N
If you’re asked to estimate
a
to 4 decimals (or have an error less than 0.0001), then solve
n
n=1
integrate and solve for N .
f (n) dn < 0.0001
N
n
(b) Alternating Series Estimation Theorem: Given a convergent alternating series
a
=
( 1)
b
,
n
n
n=1
n=1
then
R
b
N
N +1
14. Power Series
(a) A power series centered at 0
the radius of convergence is R = 0.
This happens when the Ratio Test gives
n
2
3
c
x
= c
+ c
x + c
x
+ c
x
+ ...
n
0
1
2
3
L > 1.
n=0
(b) A power series centered at a
ii. The series converges for all x. The in-
n
2
3
c
(x a)
= c
+c
(x a)+c
(x a)
+c
(x a)
+...
n
0
1
2
3
terval of convergence is (
,
) and
n=0
the radius of convergence is R =
.
(c) Radius and Interval of Convergence:
This happens when the Ratio Test gives
Perform the Ratio Test or Root Test (usu-
L = 0.
ally Ratio Test) For a given a power series
iii. The series converges on an interval (a
n
c
(x
a)
, there are only three possibil-
n
n=1
R, a+R). This happens when the Ratio
ities:
Test gives K x
a , where you need to
i. The series converges only when x = a.
solve K x
a < 1.
The interval of convergence is a and
15. Writing Functions as a Power Series
n
2
3
Given f (x) =
c
(x
a)
= c
+ c
(x
a) + c
(x
a)
+ c
(x
a)
+ ... with a radius of conver-
n
0
1
2
3
n=0
gence R > 0, then
n 1
2
3
f (x) =
nc
(x
a)
= 1c
+ 2c
(x
a) + 3c
(x
a)
+ 4c
(x
a)
n
1
2
3
4
n=1
n+1
c
(x
a)
1
1
1
n
2
3
4
f (x) dx =
= c
(x
a) +
c
(x
a)
+
c
(x
a)
+
c
(x
a)
+ ...
0
1
2
3
n + 1
2
3
4
n=0
1
You use this technique if you’re asked to find a power series of a function like
, ln(1 + x),
2
(1 + x)
1
tan
x, etc. Your goal is to differentiate or integrate your function f (x) until it’s in the proper form
A
1
u
15
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