Question
Suppose that $\sum a_{n}$ and $\Sigma b_{n}$ are series with positive terms. Prove that if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\infty$ and $\Sigma b_{n}$ diverges, $\Sigma a_{n}$ also diverges.
Step 1
This implies that there exists a positive integer $N$ such that for all $n \geq N$, $\frac{a_{n}}{b_{n}} > 1$. Show more…
Show all steps
Your feedback will help us improve your experience
Linh Vu and 58 other Calculus 2 / BC educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Proof Suppose that $\Sigma a_{n}$ and $\Sigma b_{n}$ are series with positive terms. Prove that if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\infty$ and $\Sigma b_{n}$ diverges, then $\Sigma a_{n}$ also diverges.
Infinite Series
Comparisons of Series
Suppose that $\sum a_{n}$ and $\sum b_{n}$ are series with positive terms. Prove that if $\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0$ and $\Sigma b_{n}$ converges, $\Sigma a_{n}$ also converges.
Prove that if $\Sigma a_{n}$ diverges and $\Sigma b_{n}$ converges, then $\Sigma\left(a_{n}+b_{n}\right)$ diverges.
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD