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Determine whether the series k=0-3k+14k-2converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series k=0-3k+14k-2converges to -1927.

Step by step solution

01

Step 1. Given information. 

Given a series k=0-3k+14k-2.

02

Step 2. Find if the series converges or not.

The series k=0-3k+14k-2can be expressed as k=0-48-34k.

The series role="math" localid="1648977600459" k=0-3k+14k-2is in the standard form k=0crkfor a geometric series with c=-48and r=-34.

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

Sincer=-34, it follows that the series converges.

03

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

If the geometric series k=0-3k+14k-2converges, it converges to c1-r.

So, the series k=0-3k+14k-2converges to -481--34, that is -1927.

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