Answer to Question #262688 in Calculus for Sayem

Question

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

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

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

Answer 1

use the Leibniz sign.

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

if $\alpha<1$ then the row converges

$lim_{k \to \text{\oe}} {\alpha}^{k^k}/2=\text {\oe}$

If $\alpha>1$ then the row diverges.

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

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

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

If $0< \alpha<1$

the series converges absolutely

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

If $\alpha<1$

then the row converges absolutely

$lim_{k \to \text{\oe}} {\alpha}^{k}/k=\text {\oe}$

If $\alpha>1$ then the row diverges