Solution Found!
?(a) Suppose that \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms
Chapter 10, Problem 48(choose chapter or problem)
(a) Suppose that \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\Sigma b_{n}\) is convergent. Prove that if
\(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\)
then \(\Sigma a_{n}\) is also convergent.
(b) Use part (a) to show that the series converges.
(i) \(\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}}\)
(ii) \( \sum_{n=1}^{\infty}\left(1-\cos \frac{1}{n^{2}}\right)\)
Questions & Answers
QUESTION:
(a) Suppose that \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\Sigma b_{n}\) is convergent. Prove that if
\(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\)
then \(\Sigma a_{n}\) is also convergent.
(b) Use part (a) to show that the series converges.
(i) \(\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}}\)
(ii) \( \sum_{n=1}^{\infty}\left(1-\cos \frac{1}{n^{2}}\right)\)
ANSWER:Step 1 of 4
(a) Given: 
