Series. Determine if the series converges or diverges.

| August 31, 2017

Question
Math 1452-010
11/28/2015
Written HW #2 (Extra Credit)
Due Date: Wednesday, December 9th at 4:30 PM (i.e. due at the final exam)
Directions: Please organize your work, write legibly, and solve the problems carefully. Include each major step
and justify your work as much as you reasonably can. This assignment is worth a maximum of 50 points to the
written homework category.
1. Series. Determine if the series converges or diverges. If the series converges, determine if it converges
absolutely or conditionally. If the series diverges, choose and justify the most appropriate answer from the
following three choices: ∞, −∞, or . (5 points each)
(a)
(b)
(c)



(−1) +1
sin()
sin2 ()

∑ 1.001

ln⁡ )
(

k!
=1

=1

=0

2. Power Series. Determine the convergence set and specify any points where the convergence is
conditional. (5 points each)
(a)
(b)
(c)


∑ 2
∑ !

ln()
=1

=0

=1

3. Taylor/Maclaurin Series. (5 points each)
(a) Write the Maclaurin series for () = cos2() using sigma notation. (Hint: Use the identity cos2 () =
1+cos(2)
)
2

(b) Write the Taylor series for () = ( − 1) at = 1 using sigma notation. (Hint: First find the Taylor
series for at = 1)
4. Vectors. (5 points each)
(a) True/False. For vectors ⃗, in ℝ3 , we have ⃗ ∙ = 0 if and only if ⃗ = 0 or = 0, where the dot means
dot product. Explain your answer. (See section 9.3 for the definition of the dot product.)
(b) Give an example which demonstrates that the cross product of vectors in ℝ3 is not associative. (See
section 9.4 for the definition of the cross

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