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

Determine whether the series n=1-1n+13nconverges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series n=1-1n+13n converges to 14.

Step by step solution

01

Step 1. Given information.

Given a series n=1-1n+13n.

02

Step 2. Find if the series converges or not.

The index starts with 1, rather than 0.

Note that the convergence of a series depends not upon the first few terms but only upon the tail of the series.

The standard form of geometric series is k=0crk.

Here, the series n=1-1n+13nhas c=13and r=-13.

The geometric series converges if and only if r<1.

Since r=-13, it follows that the series localid="1648883874265" n=1-1n+13nconverges.

03

Step 3. Find the value to which the series converges.

If the geometric series k=0crkconverges, it converges to c1-r.

So, the series n=1-1n+13nconverges to 131--13, that is 14.

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