Question 1

Find the sum of the series   .

Accepted Answer

A)    
B)    
C)    
D)    
E)  $\infty$ (series diverges) 
A)    
B)    
C)    
D)    
E)  $\infty$ (series diverges)

Question 2

Find the Taylor series of f(x) =   + 2x + 2 about x = -1. For what values of x does it converge to f(x)?

Accepted Answer

A)  f(x) = 2 + ,   for all x 
B)  f(x) = 1 + ,   for all x 
C)  f(x) = 1 - ,   for all x 
D)  f(x) = 1 + ,   for all x 
E)  f(x) = 1 +   , -2 < x < 0 
A)  f(x) = 2 + ,   for all x 
B)  f(x) = 1 + ,   for all x 
C)  f(x) = 1 - ,   for all x 
D)  f(x) = 1 + ,   for all x 
E)  f(x) = 1 +   , -2 < x < 0

Question 3

The series   converges.

Accepted Answer

A) True 
 B)False

Question 4

Give an upper bound for the error when e =   is approximated by the partial sum   . How large should n be taken to ensure that the error is less than 0.001?

Accepted Answer

A)    , n = 7 
B)    , n = 8 
C)    , n = 7 
D)    , n = 8 
E)    , n = 6 
A)    , n = 7 
B)    , n = 8 
C)    , n = 7 
D)    , n = 8 
E)    , n = 6

Question 5

Find the Fourier series of f(t) =   , f(t + 2π) = f(t).
Hint : To significantly simplify calculations, sketch the graph of f(t).

Accepted Answer

From the graph (shown below), it is imme

Find the sum of the series .

Find the Taylor series of f(x) = + 2x + 2 about x = -1. For what values of x does it converge to f(x)?

The series converges.

Give an upper bound for the error when e = is approximated by the partial sum . How large should n be taken to ensure that the error is less than 0.001?

Find the Fourier series of f(t) = , f(t + 2π) = f(t). Hint : To significantly simplify calculations, sketch the graph of f(t).

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Exam 10: Sequences, Series, and Power Series