does the series (sum from 5 to infinity ) 6^n/7^n converge or diverge? calculus 2, based on convergence tests

Expert Answers

An illustration of the letter 'A' in a speech bubbles

I will be using the ratio test to prove `sum_(n=1)^oo6^n/7^n`

is absolutely convergent => the serie converge =>`sum_(n=5)^oo6^n/7^n`

also converge.

Let's rewrite the serie first. `sum_(n=5)^oo6^n/7^n=sum_(n=5)^oo(6/7)^n`

Please note that in the convergence test we look at the limit of the absolute value of the ratio of the n+1 term to the n term. But sinve both of our terms are positive, we will not need them.

`lim_(n->oo)abs(a_(n+1)/a_n)=`

`lim_(n->oo)[(6/7)^(n+1)]/[(6/7)^n]=lim_(n->oo)(6/7)=6/7<1`

Hence the serie absolutely converge => the serie converge.

 

 

 

 

Posted on

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial