Power Series

Question 1

What are the three cases of turning a function into a power series? And what is home base ?

Answer 1

1) similar to home base so manipulate home base: 1/1-x

2) exponent on the denominator so we must find the simple home-base form and its power series, differentiate the form and its power series, manipulate to the original function

3) ln case, so we must take the derivative, then find the power series, then take the integral of the power series

Question 2

What happens to all the terms that only have n values and no x values when differentiating or integrating?

Answer 2

They stay the SAME

Question 3

what happens to the R as you differentiate or integrate

Answer 3

It’s preserved

Question 4

Describe the process of testing the convergence of a power series

Answer 4

Do the ratio test, find R, test the endpoints if necessary

Question 5

If the ratio test yields “0” what is concluded

Answer 5

The series converges for all x and R=infinity

Question 6

If the ratio test yields “infinity” what is concluded

Answer 6

The series converges for the center and diverges everywhere else. R=0 because there is only one value

Question 7

If the ratio test yields yields a number other than 0,1,or infinity what can be concluded?

Answer 7

There are endpoints in the form -R+c<x<R+c

Question 8

What can you do if there is a n factorial that is hard to factor out in the root test?

Answer 8

Remember 8! = 876!

Question 9

What tests can we use for endpoints convergence ?

Answer 9

• AST
• DIVERGENCE TEST
• P SERIES
• COMPARISON TEST

Question 10

Is ln(n) < n?

Answer 10

YES