Answer to Question #262688 in Calculus for Sayem

Question #262688

Determine for which value of α\alpha the given series converge absolutely, converge conditionally, or diverges.

  1. k=1((1)kα\displaystyle\sum_{k=1}^ ∞ ((-1)^k* \alpha ^(k)k)/2
  2. k=1((1)kα\displaystyle\sum_{k=1}^ ∞ ((-1)^k* \alpha k)/k

Can you kindly explain a little bit so that I can try to solve other problems. Thank you so much.


1
Expert's answer
2021-11-09T10:11:11-0500

use the Leibniz sign.

1. limkœαkk/2=0lim_{k \to \text{\oe}} {\alpha}^{k^k}/2=0

if α<1\alpha<1 then the row converges

limkœαkk/2=œlim_{k \to \text{\oe}} {\alpha}^{k^k}/2=\text {\oe}

If α>1\alpha>1 then the row diverges.


limkœαkk/2=œlim_{k \to \text{\oe}} |{\alpha}^{k^k}/2|=\text {\oe}

If α<0\alpha<0 the series converges conditionally.

limkœαkk/2=0lim_{k \to \text{\oe}} |{\alpha}^{k^k}/2|=0

If 0<α<10< \alpha<1

the series converges absolutely

2. limkœαk/k=0lim_{k \to \text{\oe}} {\alpha}^{k}/k=0

If α<1\alpha<1

then the row converges absolutely


limkœαk/k=œlim_{k \to \text{\oe}} {\alpha}^{k}/k=\text {\oe}

If α>1\alpha>1 then the row diverges



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment