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Written Homework 6, Due March 14
Written homework is to be handwritten and handed in at the beginning of discussion. In extenuating
circumstances you can notify your Teaching Assistant via email before discussion, and homework can
be submitted via alternative means the same day of discussion, under agreement with your Teaching
Assistant. Otherwise the homework will be considered late. Show all your work, correct answers
without supporting work will not receive full credit.
Question 1 (7 points)
The series X∞
k=1
kxk
5
k may or may not converge, depending on what x is. Let’s figure out for which x it
converges. This is a preview of Chapter 11.
(a) (2 points) What is the ratio ak+1
ak
? Simplify it as much as possible. Your answer should be in terms
of both x and k.
(b) (2 points) What is the limit L = lim
k→∞
|
ak+1
ak
| ?
(c) (3 points) The ratio test now tells you that the series converges if the number from your answer to
part (c) is less than 1. For what values of x is it true that L < 1? What can you conclude about
the series for these values of x?
Page 1
Written Homework 6, Due March 14
Question 2 (6 points) Determine if the series X∞
k=0
(−1)k
k
1 + k
3
is absolutely convergent, conditionally
convergent or divergent. Show all your work!
Page 2
Written Homework 6, Due March 14
Question 3 (6 points) Determine if the series X∞
k=10
(−1)k
√
k − 3
is absolutely convergent, conditionally
convergent or divergent. Show all your work!
Page 3
