Which statement is true for the following series? ∑k=1∞(−1)k5k+6 1. Cannot be determined 2. Divergent 3. Conditionally convergent 4. Absolutely convergent

Question

Which statement is true for the following series?   ∑k=1∞(−1)k5k+6  1. Cannot be determined  2. Divergent  3. Conditionally convergent   4. Absolutely convergent

Elaine Verrett · Accepted Answer

Given that  ∑k=1∞(−1)k5k+6  A series ∑n=1∞an is called absolutely Convergent if  ∑n=1∞|an| is Convergen.  And if ∑n=1∞|an| is divergent but ∑n=1∞an is Convergent.  Here ∑n=1∞an is called Conditionally Convergent.  Here ak=(−1)k5k+6  ∑k=1∞|an|=∑k=1∞15k+6  Limit compoison test.  if ∑n=1∞an and ∑n=1∞bn are two series and let limanbn=1 non zero finite number  then ∑n=1∞an and ∑n=1∞bn both Convergent or both divergent  ∴ let bk=1k  ∴limk⇒bkqkbk=lim15k+61k  ⇒limk⇒bkqkbk=limk⇒bkk5k+6=15≠0 finite number  ⇒∑k=1∞1k and ∑k=1∞15k+6 both conv. or div.  Since ∑k=1∞1k is divergent therefore ∑k=1∞15k+6 is divergent by limit Compoison test  ⇒∑k=1∞(−1)k5k+6 can not be absolutely Convergent.  Now. Leibnitz series test.

Rita Miller · Accepted Answer

Is it all solution?

RizerMix · Accepted Answer

Sure! Very detailed description!

Which statement is true for the following series? \sum_{k=1}^\infty \frac{(-1)^k}{5k+6} 1.

Answered question

Answer & Explanation

New Questions in Calculus 2