1. Sequences and Series & Taylor series
Patrick Doyle
Elwin Martin
Charles Ye

2. Series and Sequences
Sequence: function whose domain is a set of all positive integers (n=1,2,3…) 2

3. Limit of a Series
If the sequence converges, then

4. example
Series diverges
Series converges

5. nth term test
If   o or if the limit does not exist, then the series diverges.
Apples to any series
Cannot be used to prove convergence

6. example
Not enough information
Series diverges

7. Geometric series Converges if the absolute value of r is less than one
Diverges if the absolute value of r is greater than or equal to one
Where a is the first time

8. example

9. Alternating harmonic series

10. Harmonic series
Always diverges

11. Telescoping series Series where the later terms start to cancel out
Converges to the numbers that do not cancel out

12. example
Converges to 1

13. Integral Test
Converges only if converges Diverges only if diverges Where the function is continuous, positive, and decreasing

14. example

15. P-series test If p is greater than one, the series converges
If p is equal or less than one, the series diverges

16. Direct Comparison Test
Converges if and converges
Diverges if and diverges

17. example

18. Limit Comparison Test
If is a number greater than zero and converges, then
converges
If is a number greater than zero and diverges, then
diverges
-Both series a and b must be greater than zero
- The limit must not be infinity, zero, or negative

19. Root Test
There exists r such that if r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

20. example

21. Ratio Test
There exists a number r such that: if r < 1, then the series converges; If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

22. Error in Alternating series

23. Radius and Interval of Convergence
In some instances there are series written in terms of x such as and in such cases there are a range of x values for which the series converges.
The radius of curvature and interval of convergence are found by applying the ratio test and setting the ratio to less than one and solving for x.
Once the interval has been found, check the end points.

24. example

25. Power Series
Representation of a function as a summation of infinitely many polynomials.
C= x value at which the series is centered

26. Finding the Coefficient
I FOUND IT!!!
Using this, you can plug it in for several terms then find a the equation for the general term!
Now you have a Taylor Series!

27. (Actually finding the coefficient)

28. Example
Find the first 3 terms and the general term of the Taylor series for sin(x) centered at 0.
f (x) = sin(x)
f (0) = 0
f '(x) = cos(x)
f '(0) = 1
f '' = -sin(x)
f ''(0) = 0
f ''' = -cos (x)
f '''(0) = -1
f '''' = sin(x)
f ''''(0) = 0
f''''‘=cos(x) f''''‘(0)=1

29. Manipulating the Series
You can tweak the formulas you memorized to make finding the Maclaurin Series for many more equations much faster and EASIER. So do that.
You can multiply the whole series by a number, substitute for x, or take the integral or derivative.

30. Maclaurin Series
A Maclaurin series is a Taylor Series that centers at 0 (i.e. c=0).
Here are some to memorize:

31. Example
Find the first four terms of the Maclaurin series for 5e2x.

33. Lagrange Error Bound
Where f(x) is centered at c and a is some number between x
and c.