Sequences and Series

1. Sequences and Series Andrew Taylor Alec Seco

2. Basic Sequences • Sequence: a function whose domain is a set of positive integers • Ex. {1,1/2,1/4,1/8,1/16} or (an = ( 1/2 )n-1) • Either converges or diverges • converge: sequence approaches some number L • diverge: sequence doesn’t approach any finite number or oscillates

3. Convergence/Divergence • To test, simply solve • Ex. diverge or converge? • Ex. diverge or converge? converges diverges due to oscillation • Sometimes you might have to use L’Hopital’s rule

4. Geometric Sequences • Involves multiplying each term by a fixed term • Generally in the form ratio Rule for Geometric Convergence/Divergence |r| < 1  sequence converges |r| 1  sequence diverges

5. Arithmetic Sequence • Each term involves adding/subtracting a number to the one before • Ex: {1,3,5,7,9,11} • ALL arithmetic sequences diverge except when the added term is 0 • {0,0,0,0} or {3,3,3,3} (duh.)

6. SERIES • Series basically take sequences and add up the numbers • Again, we look at convergence and divergence • General format for series • Evaluates the sum of a sequence from 1st (or 0th) term to any number • Infinite series- what we’ll mostly look at concerning convergence and divergence; has unlimited number of terms • Finite series – evaluate from n=1 or 0 to a certain number • Find the sum of a certain number of terms • Ex: = 1+2+3+4+5 = 15

7. Special Series - Geometric • Geometric series are sums of geometric sequences, similar format • Rules of Series convergence/divergence are the same as sequences • |r| < 1, converge • |r| > or = 1, diverge **Arithemetic Series** These never converge unless they are the zero set {0,0,0,0} (duh.)

8. Special Series: p-series • General form: • Conditions for convergence: p<1 • Conditions for divergence: p 1 • Special Case: Harmonic Series • Most common p-series • diverges ***becomes extremely useful with Direct Comparison Test***

9. Special Series: Alternating Series • A series where the sign alternates • General form: • If it meets 2 qualifications, it converges

10. Alternating Series Error • The error will always be smaller than the first neglected term • Say you found the sum for the first four terms; the error will always be less than the magnitude of the fifth term • Ex: Find the error of

11. Absolute/Conditional Convergence • Absolute convergence: just checking if the absolute value of the series converges • Can be a quick alternative before using Alt. Series Test • If a series absolutely converges, then it always converges

12. Power Series: An Introduction • General form for a Taylor Series centered at x = a: • If f(x) is centered at 0, then the power series is called a Maclaurin Series. • The series can also be written in summation notation (sigma), from above:

13. Convergence of Power Series • Given a power series centered at x=a, exactly one of the following will be true: • The series converges for all x. (-∞,∞) • The series converges for only one x=a. • The series converges on some interval (b,c)… but in this case you must also check the endpoint values b and c, assigning closed or open brackets as necessary.

14. Important Power Series • The following are series that MUST be memorized: • 1. ex - converges for all x. (-∞,∞) • 2. - converges on interval (-1,1)

15. Important Power Series • 3. sin(x) – converges for all x. (-∞,∞) • 4. cos(x) – converges for all x. (-∞,∞)

16. How To Find Convergence • 1. Apply the ratio test to the general term for the series. • 2. Set the resulting expression < 1. (Don’t forget the absolute value…) • 3. Find the interval of convergence. • 4. Check endpoints.

17. How to Use a Known Series to Write the Power Series of a Similar One • Ex. • But this is really • Which is just with x2 replacing x and then multiplying by a factor of 2.

18. How to Use a Known Series to Write the Power Series of a Similar One • So by applying those two changes to each term in the known Taylor series for we can obtain the power series for . • This gives us: • And then the answer:

19. Lagrange Error Bound: What You Need To Know to Get That One Part of That One FRQ • A typical question reads like this: • Prove that • P(x) is an nth degree Taylor polynomial for a given f(x). • X is some value to be plugged into the polynomial. • Q is the target error. • Showing that the error is less than some value. • General Formula To Use: • But what does that mean???

20. Lagrange Error Bound: What You Need To Know to Get That One Part of That One FRQ • Here’s what that formula says to do: • 1. Find , which is the derivative of f(x) one order higher than the last one you took when finding P(x). Ex. If P(x) was a third-degree Taylor polynomial, we are talking about the 4th derivative. • 2. Find the maximum value can be between the value the function is centered at and your given “x” value. • 3. Multiply this value by and by • “C” is value f(x) is centered at, so when c=0, its just the given x value to the n+1 power.