Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Considertheseriesk=1(-1)k+1akandk=1(-1)kakwhere(i){ak}isasequenceofpositivenumber(ii)thesequence{ak}isstrictlydecreasing(iii)limkak=0

A series can fail two of the three conditions, (i), (ii) and (iii), and still converge. Which of the three conditions must an alternating series pass in order to converge?

Short Answer

Expert verified

According to the Divergence Test if the sequence {ak} does not converge to zero, then the

series k=1ak, diverges.

Hence, for the seriesk=1(-1)k+1akandk=1(-1)kak, to be convergent it must pass the third condition.

Step by step solution

01

Step 1. Given 

Giventheseriesk=1(-1)k+1akandk=1(-1)kakwhere(i){ak}isasequenceofpositivenumber(ii)thesequence{ak}isstrictlydecreasing(iii)limkak=0

02

Step 2. Explanation

If the third condition is not passed by an alternating series , then limkak=0.

According to the Divergence Test if the sequence {ak} does not converge to zero, then the

series localid="1649138812346" k=1ak, diverges.

Hence, for the serieslocalid="1649138756493" k=1(-1)k+1akandk=1(-1)kak, to be convergent it must pass the third condition.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Study anywhere. Anytime. Across all devices.

Sign-up for free