Review for MATH 262
1/30/2010
Sequences
Review for MATH 262 series tests error approximation power series
Nth term test (Divergence Test)
Series
a n 0
n
Easy stuff right...?
an 0
If
If limit = 0, test INCONCLUSIVE
Telescoping Series
lim n
, series DIVERGES
Geometric Series
ax
Examples: Expand, cancel terms, and apply limit to the remaining nth terms
n
n 0
If |x|< 1, Series CONVERGES And Sum = a 1 x
Else, Series DIVERGES
Use partial fraction decomposition
Note: formula works for starting index at n=0!!
1
1/30/2010
P-Series
1 np
Integral Test
If P>1, Series CONVERGES If P=1, Series DIVERGES If P 0 then, the
Root Test lim n n
| an | r
0≤r1 Series DIVERGES R=1 Test INCONCLUSIVE
2
1/30/2010
Alternating Series Test (AST)
For series of the form: (1) n a n With an is positive If an is a decreasing sequence and, lim n
an 0
Series CONVERGES (does not tell you if it is absolutely or conditionally convergent however!!) Series DIVERGES otherwise (not implied by this test, but usually if this test fails…it sort of implies that the nth term test worked…)
If a series is absolutely convergent, then the series itself is also convergent. i.e. if | an | converges, this IMPLIES an CONVERGES as well!!
Conditional & Absolute Convergence
a CONVERGES… And | a | CONVERGES If
n
Then
BUT…if
Then
Check if the absolute value of the series converges first…if it does (i.e. series is absolutely convergent), it implies that the series itself converges too! If it does not, use the AST to check if the series is at least conditionally convergent. This saves you a step compared to if you were to use AST first, as AST only tells if your series converges or diverges and not whether it is absolutely or conditionally convergent.
Power Series
To check for interval of convergence, apply ratio test Must always check the lower and upper bounds of the interval Radius of convergence is just half the distance between the bounds (i.e. distance from the center to a bound) Methods for solving for power series representations include differentiation, integration, substitution look at assignment 3
n
CONVERGES ABSOLUTELY
| a
a
n
n
| DIVERGES
CONVERGES CONDITIONALLY
Error Approximations
Tip: For questions asking whether a series is absolutely convergent, conditionally convergent, or divergent:
a
n
For positive series
An+1 + Sn ≤ S ≤ An + Sn
Where An =
f ( x)dx
, and An+1 =
f ( x)dx
n 1
n
We can thus approximate
Satisfying | S S * | An An1 Error n
S S n* S n
An 1 An 2
2
For alternating series
|S – Sn| ≤ |an+1| ≤ Error
Ex: f(x) = 10xtan-1(4x) Look at tan-1(4x) first: Either you find the power series representation of tan-1(4x) by differentiating it, manipulating it in order to change into a power series form, integrating the power series form in order to get the power series representation of tan-1(4x) or… Apply differentiation/integration to tan -1(x) instead, and once you get the power series representation of tan-1(x), substitute 4x into x (MUCH EASIER!!)
Multiply 10x to the power series to get your final answer
3
1/30/2010
Some important Maclaurin Series you definitely should memorize…
Taylor Series
Of the form:
Where fn(a) is the nth derivative of the function f(x)
Remember: You can always rederive these using the formula on the last page
A good web site
http://tutorial.math.lamar.edu/
Helpful not only for math262 but also for the other math courses coming next
4
Sequences
Review for MATH 262 series tests error approximation power series
Nth term test (Divergence Test)
Series
a n 0
n
Easy stuff right...?
an 0
If
If limit = 0, test INCONCLUSIVE
Telescoping Series
lim n
, series DIVERGES
Geometric Series
ax
Examples: Expand, cancel terms, and apply limit to the remaining nth terms
n
n 0
If |x|< 1, Series CONVERGES And Sum = a 1 x
Else, Series DIVERGES
Use partial fraction decomposition
Note: formula works for starting index at n=0!!
1
1/30/2010
P-Series
1 np
Integral Test
If P>1, Series CONVERGES If P=1, Series DIVERGES If P 0 then, the
Root Test lim n n
| an | r
0≤r1 Series DIVERGES R=1 Test INCONCLUSIVE
2
1/30/2010
Alternating Series Test (AST)
For series of the form: (1) n a n With an is positive If an is a decreasing sequence and, lim n
an 0
Series CONVERGES (does not tell you if it is absolutely or conditionally convergent however!!) Series DIVERGES otherwise (not implied by this test, but usually if this test fails…it sort of implies that the nth term test worked…)
If a series is absolutely convergent, then the series itself is also convergent. i.e. if | an | converges, this IMPLIES an CONVERGES as well!!
Conditional & Absolute Convergence
a CONVERGES… And | a | CONVERGES If
n
Then
BUT…if
Then
Check if the absolute value of the series converges first…if it does (i.e. series is absolutely convergent), it implies that the series itself converges too! If it does not, use the AST to check if the series is at least conditionally convergent. This saves you a step compared to if you were to use AST first, as AST only tells if your series converges or diverges and not whether it is absolutely or conditionally convergent.
Power Series
To check for interval of convergence, apply ratio test Must always check the lower and upper bounds of the interval Radius of convergence is just half the distance between the bounds (i.e. distance from the center to a bound) Methods for solving for power series representations include differentiation, integration, substitution look at assignment 3
n
CONVERGES ABSOLUTELY
| a
a
n
n
| DIVERGES
CONVERGES CONDITIONALLY
Error Approximations
Tip: For questions asking whether a series is absolutely convergent, conditionally convergent, or divergent:
a
n
For positive series
An+1 + Sn ≤ S ≤ An + Sn
Where An =
f ( x)dx
, and An+1 =
f ( x)dx
n 1
n
We can thus approximate
Satisfying | S S * | An An1 Error n
S S n* S n
An 1 An 2
2
For alternating series
|S – Sn| ≤ |an+1| ≤ Error
Ex: f(x) = 10xtan-1(4x) Look at tan-1(4x) first: Either you find the power series representation of tan-1(4x) by differentiating it, manipulating it in order to change into a power series form, integrating the power series form in order to get the power series representation of tan-1(4x) or… Apply differentiation/integration to tan -1(x) instead, and once you get the power series representation of tan-1(x), substitute 4x into x (MUCH EASIER!!)
Multiply 10x to the power series to get your final answer
3
1/30/2010
Some important Maclaurin Series you definitely should memorize…
Taylor Series
Of the form:
Where fn(a) is the nth derivative of the function f(x)
Remember: You can always rederive these using the formula on the last page
A good web site
http://tutorial.math.lamar.edu/
Helpful not only for math262 but also for the other math courses coming next
4
