unit 10

Question 1

Error bounds

Answer 1

1) Encuentras partial sum de n
2) Encuentras el # term n+1 _> a(n+1)
3) Partial sum -/+ a(n+1) = error bounds

Question 2

McLaurin/taylor expansion

Answer 2

f(a) + x * f’(a) + (x^2)/2! * f’‘(a) + (x^3)/3! * f’’‘(a)….

Question 3

Lagrange error bounds

Answer 3

Rn = M|x-a|^(n+1) / (n+1)!
M = max value of derivatives at given point
a = center
n = degree

Question 4

Intervals of convergence

Answer 4

1) Do the ratio test
2) |x-c|< r
3) -r < (x-c) < r
4) c-r < x < c+r
- If ratio test limit = 0, the series converges for all numbers
- If ratio test limit = inf. the series converges at its center

Question 5

Ratio test

Answer 5

USE FOR:
|a (n+1)|
————
|a (n)|

Question 6

Derivatives of power series

Answer 6

1) Write the first terms -> x + x^2 / 2!…
2) Take the derivative of each term
——————————————————
if LC OR translation
3) Multiply everything by LC
4) Subtract from translation

Question 7

Derivatives of power series (general form)

Answer 7

• Just take the derivative of the equation
• Don’t worry about the (-1)^n or the denominator if it has an n

Question 8

Integrals of power series

Answer 8

1) Write the first terms -> x + x^2 / 2!…
2) Take the antiderivative of each term
3) Plug in bounds and simplify

Question 9

Integrals of power series (general form)

Answer 9

• Take the antiderivative of the top term
  Don’t worry about anything else

Question 10

Absolute convergence

Answer 10

• If |a (n)| converges, then a(n) converges absolutely
• If |a (n)| diverges, then a(n) diverges absolutely

Question 11

Conditional convergence

Answer 11

• If |a (n)| diverges, a(n) could converge by the alternating series test

Question 12

Limit comparison test

Answer 12

USE FOR:
1) Simplify equation to the greatest terms (as if you were to take the limit)
2)

Question 13

Geometric series test

Answer 13

USE FOR: (2/5) ^(n-1) OR 1/9^n
1) Rewrite to look like:
something * somethingElse^n
- somethingElse = r

2) Converges if 0<|r|<1

3) Diverges if |r|=>1

Question 14

Integral test

Answer 14

• ONLY IF POSITIVE, INCREASING, AND CONTINUOUS
• Look at the integral by replacing n with x, the function will behave like it
  USE FOR: