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Explain why or why not Determine whether the
Chapter 12, Problem 39E(choose chapter or problem)
39. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose that \(0<a_{k}<b_{k}\). If \(\sum a_{k}\) converges, then \(\sum b_{k}\) converges.
b. Suppose that \(0<a_{k}<b_{k}\). If \(\sum a_{k}\) diverges, then \(\sum b_{k}\) diverges.
c. Suppose \(0<b_{k}<c_{k}<a_{k}\). If \(\sum a_{k}\) converges, then \(\sum b_{k}\) and \(\sum c_{k}\) converge.
Questions & Answers
QUESTION:
39. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose that \(0<a_{k}<b_{k}\). If \(\sum a_{k}\) converges, then \(\sum b_{k}\) converges.
b. Suppose that \(0<a_{k}<b_{k}\). If \(\sum a_{k}\) diverges, then \(\sum b_{k}\) diverges.
c. Suppose \(0<b_{k}<c_{k}<a_{k}\). If \(\sum a_{k}\) converges, then \(\sum b_{k}\) and \(\sum c_{k}\) converge.
ANSWER:Problem 39E
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose that 0 <ak <bk . If ∑ak converges, then ∑bk converges.
b. Suppose that 0 <ak <bk . If ∑ak diverges, then ∑bk diverges.
c. Suppose 0 < bk <ck < ak . If ∑ak converges, then ∑bk and ∑ck converge.
Solution:
Step 1
a. Suppose that 0 <ak <bk . If ∑ak converges, then ∑bk converges.
Consider the partial sums and
If 0 <ak <bk , then
Taking limits, we get
So, if converges it does not proves that
converges too.
Instead, if it was given that converges then it is true that
will also converge.