BC Calculus
Chapter 9 Review
I. Convergence/Divergence Tests

Special Series o P-series, Geometric, Alternating, Telescoping

Ratio, Root, nth term, Integral, Basic/Direct Comparison, Limit Comparison

Absolute/Conditional Convergence
II. Sums of Series

Partial Sums

Geometric Sum

Telescoping Sum
III. Interval and Radius of Convergence

Note: usually use Geometric, Ratio, or Root test
IV. Taylor/Maclaurin Series

Special Maclaurin series

Taylor series centered at x = a
 Taylor’s Theorem/Lagrange Error Bound
Problems:
1. Find the radius of convergence, the interval of convergence, and the values of x for which the series converges absolutely and conditionally. a.) n



0
(
 x ) n n !
b.) n



0
2
3 n
( x

1 ) n
2. Find the Maclaurin series for the functions below. c.) n



1 x n n n d.) n



1 n !
2 n x
2 n a.) f ( x )

1

1
6 x b.) f ( x )
  x
 sin x c.) f ( x )
  xe
 x
2
3. Find the first four nonzero terms and the general term of the Taylor series. d.) a.) f ( x )

1
3
 x
around a = 2 b.) f ( x )
 x
3 
2 x
2 f ( x )
 ln( 1

2 x )

5 around a = –1 c.) f ( x )

1
around a = 3 d.) f ( x )
 sin x around a =
 x
4. Determine if the series converges absolutely, converges conditionally, or diverges. Give reasons for your answers. a.) n



1
(

1 ) n n b.) n



1 ln n n
3 c.) n



2 n (ln
1 n )
2 d.) n



1
(

1 ) n
( n
2
2 n
2 

1 ) n

1 e.) n



1
2 n
3 n n n f.) n



1
1 n ( n

1 )( n

2 )

7.
5. Find the sum of the series. a.) n



3
4 n
2 
1
8 n

3 b.) n



1

4

5
7

 n

1
6. Let P
4
( x )

7

3 ( x

4 )

5 ( x

4 )
2 
2 ( x

4 )
3 
6 ( x

4 )
4 be the Taylor polynomial of order 4 for the function f at x = 4 . Assume f has derivatives for all orders for all real numbers. a.) Find f(4) and f’’’(4)
. b.) Write the second order Taylor polynomial for f’
at x = 4 and use it to approximate f’(4.3)
. c.) Write the fourth order Taylor polynomial for g ( x )
 x

4 f ( t ) dt at x = 4 . d.) Can the exact value of f(3) be determined from the information given? Justify your answer.

Answers to problems
1. a.) ROC = ∞
IOC = all real #
Converges absolutely for all real #
Converges conditionally none (since converges absolutely for all real #) c.) ROC = ∞
IOC = all real #
Converges absolutely for all real #
Converges conditionally none (since converges absolutely for all real #) b.) ROC = 3
IOC = [–7, –1]
Conv. abs. (–7, –1)
 notice not at endpts.
Conv. cond. At x = –7 d.) ROC = 0
IOC = x = 0 only
Conv. abs. x = 0
Conv. cond. None
2. a.) n



0
( 6 n x ) b.) n



1
(

1 ) n
( 2 x
2 n

1 n

1 )!
c.) n



0
(

1 ) n x
2 n

1 n !
3. a.) 1

1
c.)
3
d.)


( x
( x
1
9

( x
 
2 )


3 )
)

( x

1
3 !
( x

1
27
2 )
2
( x
 
)
3

...


3 )
2

( x

1
5 !
( x
1
81
( x


2 ) n

)
5

3 )
3


...

1
7 !
( x
 b.)
(

1 ) n
2

7 ( x
 x

3
 n

3 n

1
1 )

)
7 
...

(

1 ) n

1

5 ( x

1 )
2
( x

( 2 n

)
2 n

1

1 )!
d.)

( x

1 )
3 n



1
(

1 )
( 2 x ) n

...

0 n
4. a.) Conditionally convergent (Alternating series test…does not converge absolutely because of p-series test)
b.) Absolutely convergent (Direct Comparison Test with 1/n 2 )
c.) Absolutely convergent (Integral Test)
d.) Divergent (n th term test)
e.) Absolutely convergent (Root Test or Ratio Test)
f.) Absolutely convergent (Direct comparison test with 1/n 3/2 )
5. a.) S = 1/6 b.) S = 14 f ' ' ' ( 4 )
6. a.) Since the constant term is f(4) , f(4) = 7 . Since
 
2 , f’’’(4) = –12 .
3 !
b.) (Take the derivative of the polynomial given) f ' ( x )
 
3

10 ( x

4 )

6 ( x

4 )
2 
24 ( x

4 )
3
. f'(4.3) = –0.54
c.) (Integrate term by term) g ( x )

7 ( x

4 )

3
2
( x

4 )
2
7.

5
3
( x

4 )
3 
1
2
( x

4 )
4