Question
Prove: If $\sum_{k=1}^{\infty} a_{k}$ diverges, so does $\sum_{k=1}^{\infty} c a_{k}$ for $c \neq 0$.
Step 1
Step 1: Assume that the series $\sum_{k=1}^{\infty} c a_{k}$ converges. Show more…
Show all steps
Your feedback will help us improve your experience
Vipender Yadav and 79 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
Show that $$\sum_{k=1}^{\infty}\left(\frac{k+1}{k}\right)^{k} \text { diverges. }$$
Infinite Series
(a) Prove that if $\sum_{k=0}^{\infty} a_{k}$ is a convergent series with all terms nonzero, then $\sum_{k=0}^{\infty}\left(1 / a_{k}\right)$ diverges. (b) Suppose that $a_{k}>0$ for all $k$ and $\sum_{k=0}^{\infty} a_{k}$ diverges. Show by example that $\sum_{k=0}^{\infty}\left(1 / a_{k}\right)$ may converge and it may diverge.
(a) Prove that if $\sum_{k=0}^{\infty} a_{k}$ is a convergent series with all terms nonzero, then $\sum_{x=0}^{\infty}\left(1 / a_{k}\right)$ diverges. (b) Suppose that $a_{k}>0$ for all $k$ and $\sum_{k=1}^{\infty} a_{k}$ diverges. Show by example that $\sum_{k=0}^{\infty}\left(1 / a_{k}\right)$ may converge and it may diverge.
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD