1. Convergence, Series, and Taylor Series
By: Ashwin, Sean, and Adam

2. 52. Convergence/Divergence of Sequences
Find the limit of the sequence as it approaches infinity
If the limit approaches infinity, then the sequence diverges
If the limit approaches a finite value, then the sequence converges

3. 53. Nth Term Test Can only be used to test divergence
If limit equals 0, then you must use a different test

4. 54. Geometric Series
Series diverges if absolute value of r is greater or equal to 1
Series converges to sum of a/(1-r) if absolute value of r is between 0 and 1
A1 is first term and r is rate

5. 55. P Series Test and Integral Test
Series diverges if exponent p is less than or equal to 1
Series converges if p is greater than 1
Integral Test
Series converges if integral converges
Series diverges if integral diverges
*function must be positive, continuous, and decreasing on interval 1 to infinity

6. 56. Direct Comparison Test
Compare original function to a simple function
Then determine whether the comparison function is bigger or smaller than the original function
If the smaller function diverges or bigger function converges, then the other function does the same
SDBC
Students dominate BC

7. 57. Limit Comparison Test
Bn is the comparison function
An is the original function
Take the limit of the original function over the comparison function
If the limit equals a positive finite number, then the original function acts the same as the comparison function
If comparison function diverges, then so does the original function
If comparison function converges, then so does the original function

8. 58. Alternating Series Test
3 conditions must be met
1. series alternates in sign
2. series is decreasing
3. limit as n approaches infinity of the series equals 0

9. 59. Ratio Test
Take the limit of the absolute value of (An+1/An) as n approaches infinity
If the limit is less than 1 then the series converges absolutely
If the limit is greater than 1, then the series diverges
If the limit equals 1, then the test in inconclusive

10. 60. Root Test
Take the limit an n approaches infinity of the nth root of the series
If the limit is less than 1, then the series is absolutely convergent
If the limit is greater than 1, then the series is divergent
If the limit equals 1, then the series is inconclusive

11. 61. Absolute vs Conditional Convergence
series is absolutely convergent if absolute value of series converges
If absolute value of series diverges and the original series converges, then the series is conditionally convergent

12. 62. Center, Interval of Convergence, Radius of Convergence
Use the ratio test of the absolute value of the series to find the expression for the interval of convergence
Set the expression less than 1 and greater than -1 to find the open interval of x
Check endpoints to see if they’re inclusive or exclusive by using a series test
If IOC is a single value, then ROC is 0
If IOC is a range of finite values, then ROC is the average of the difference of the values
If IOC is all real numbers, then ROC is infinity

13. 63. Finding a function’s power series using a/(1-r)
Must get the power series to the form a/(1-r)
Manipulate expression to get the expression centered at c
Once you have the function in the form a/(1-r), you can rewrite it as a power series by taking “a” and multiplying it by “r”
To find the interval of convergence, set “r” less than one and greater than -1 and solve for x
No need to check endpoints

14. 64. Derivatives and Integrals of Series
Take the derivative or integral of the term with “x” using the power rule
Find IOC by plugging in endpoints found by using ratio test on original series to see if the new series converges or diverges at that point
All other terms with “n” are constants
Don’t take derivative or integral of these

15. 65. Creating a Taylor Series
Plug into formula
Easier to memorize pattern
C = center
(Given in problem)

16. 66. 4 known McLaurin Series
Sin x
Cos x
e^x
1/(1-x)
e^x

17. 67. Alternating Series Error
Series must satisfy conditions of an alternating series
Lim as n approaches infinity equals zero
Series is decreasing
States that error will be less than or equal to the absolute value of the next unused term

18. 68. LaGrange Error Bound
((x-c)^n+1)/(n+1)! Represents the first unused term in the Taylor series
Multiplying this by the max gives a safe upper bound for the error

19. Homework What we did in class Convergence of series
9, 11, 13, 15-17, 19-23
Taylor Series
1, 5, 8-12
What we did in class
Convergence of Series
1-8,10,12,14, 18
Taylor Series
2,3,4,6, 7