Calc 2 Ch 8 Test

Front 1

Series

Back 1

An infinite set of numbers being added together

Front 2

Geometric series

Back 2

a - coefficient, r - ratio

Hint 2

Front 3

What happens when r (ratio) of a series is between -1 and 1? ( -1 < r 1 )

Back 3

The limit of the sequence is 0 and the series is a/(1-r)

Front 4

What happens If r > 1 or r <= -1?

Back 4

The series and sequence is divergent

Front 5

Example 1 of Series Problem

Back 5

Front 6

Example 2 of Series problem

Back 6

Front 7

How to find the sum of a geometric series

Back 7

1) identify it as geometric, usually all terms will have an exponent of n

2) find the first term (a) and the common ratio (r)**, the ratio is the leading nth terms of the problem

3) if the absolute value of the common ratio is greater than 1 it will diverge, if less than 1 it will converge at a point

4) find the point it converges with the formula : a/(1 - r)

**dont simplify common ratio fraction

Front 8

What makes a series absolutely convergent?

Back 8

If a series of absolute values are convergent

Front 9

(ratio test) what happens If lim n-> inf |a(n+1)/a(n) | = L < 1?

Back 9

The series is absolutely convergent.

Front 10

(ratio test) what happens If lim n-> inf |a(n+1)/a(n) | = L > 1?

Back 10

The series is divergent

Front 11

(ratio test) what happens If lim n-> inf |a(n+1)/a(n) | = 1?

Back 11

Ratio test is inconclusive, could be convergent or divergent

Front 12

Example 1 of convergence problem

Back 12

Front 13

Example 2 of Convergence problem

Back 13

Front 14

Example 3 of Convergence problem

Back 14

Front 15

How do you find the limit of a sequence?

Back 15

Divide all the terms with the leading terms then evaluate what happens when it goes to infinity.

Front 16

What do you do for series equations that have 2 terms over one?

Back 16

Break it into 2 series problems, evaluate both sums separately then add the two sums together at the end

Front 17

How do you find the interval of x in a series problem?

Back 17

1) find the first term (a) and the common ratio (r)

2) set up equation for |r| < 1 once you have found r and solve for it.

Front 18

How do you solve a convergence problem?

Back 18

Use the ratio test

Front 19

What do you do if the ratio test is inconclusive?

Back 19

Find the sum by dividing all terms by the leading term in the denominator and evaluating it to infinity.

Front 20

What is a power series?

Back 20

A series with this form

Front 21

What do you find in a power series problem?

Back 21

The interval of convergence and radius of convergence.

Front 22

Example 1 of Power Series problem

Back 22

Front 23

Example 2 of Power Series Problem

Back 23

Front 24

Example 3 of a power series problem

Back 24

Front 25

How do you solve a power series problem?

Back 25

1) Do the ratio test and separate the coefficients and x terms from the n terms

2) If the limits taken at the n term part of the problem evaluate to one then the radius of convergence is the inverse of the coefficient of x.

3) set up the inequality -1 < (x terms) < 1 and solve for x to get the interval of convergence.

Front 26

What is a partial sum for a power series? (Section 8.4)

Back 26

A specific interval of n in the power series that creates an approximation of the function used.

Front 27

What is the radius of convergence for a power series?

Back 27

Its half the distance of the interval of convergence

Front 28

What form does the function have to be in to be represented as a power series?

Back 28

Front 29

Example 1 of representing functions as power series

Back 29

Front 30

Example 2 of representing a function as a power series

Back 30

Front 31

Example 3 of representing a function as a power series

Back 31

Front 32

How do you solve a representing a function as a power series problem?

Back 32

1) put the function into the form: 1/(1-x). You may have to modify the function by pulling a term or constant and/or manipulating x

2) you can then change the form to x^n (distribute n to the coefficients of x)

3) use this form to find the amount of terms specified

4) find the radius of convergence by setting up an inequality problem |x|< 1. This will give you the interval and the radius is half the size of the interval

Front 33

What is a taylor polynomial?

Back 33

One that follows the form (f^n(x))/n!

Front 34

What is the maclaurin polynomial for sin(u)?

Back 34

u - (1/3!) u ^3 + (1/5!) u ^5 - (1/7!) u ^ 7

Front 35

What is the Maclaurin series for e^x?

Back 35

1 + x + x^2 / 2! + x^3 / 3!

Front 36

What is the expression for the series : ((-1)^n(1/3)^(2n+1)) / (2n+1)

Back 36

Arctan(1/3)

Front 37

What is the expression for the series ((-1)^n(1/3)^(2n))/(2n)!

Back 37

Cos(1/3)

Front 38

What is the expression for the series: (1/3)^n/n!

Back 38

E^(1/3)

Front 39

What is the expression for this series: ((-1)^n(1/3)^(2n+1))/(2n+1)!

Back 39

Sin(1/3)

Front 40

What is the pattern of a maclaurin series?

Back 40

f(0) + f'(0)x + (f"(0)x^2) / 2! + (f"'(0)x^3) / 3! + ....

Front 41

What is a taylor series?

Back 41

(F^n(a) / n!) (x - a)^n

Front 42

What is a binomial series?

Back 42

Front 43

Example 1 of maclaurin problem

Back 43

Front 44

Example 2 of maclaurin problem

Back 44

Front 45

Example 3 of maclaurin problem

Back 45

Front 46

How to solve a maclaurin problem?

Back 46

1) memorize the known maclaurin like those for sin and cos, otherwise find the derivatives of the for the amount of terms asked for.

2) once you have the derivative, plug in 0 into all the terms.

3) take the results and plug them into the series : f(0) + f'(0)x + (f"(0)x^2) / 2! + (f"'(0)x^3) / 3! + ...

Front 47

Example 1 of a taylor problem

Back 47

Front 48

Example 2 of a taylor problem

Back 48

Front 49

Example 3 of taylor problem

Back 49

Front 50

How do you solve a taylor problem?

Back 50

1) find the amount of derivatives requested

2) place in the given value for a for each of the derivatives.

3) treat each result as the coefficient for the corresponding term in the format : (f^n(a))/ n!)

Front 51

Example of a binomial problem

Back 51

Front 52

How do you solve a binomial problem?

Back 52

1) rearrange the function so it has the format (1 + x)^k

2) equate it to the series: 1 + kx + (k(k-1)/2!) x^2 + (k(k-1)(k-2)/3!)x^3 + ...

Front 53

What are all the topics in ch. 8?

Back 53

- Series

- Other Convergence Tests

- Power Series

- Representing Functions as Power Series

- Taylor and Maclaurin Series