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Determine whether the series k=04k+132kconverges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series k=04k+132k converges to 365.

Step by step solution

01

Step 1. Given information.

Given a series k=04k+132k.

02

Step 2. Find if the series converges or not.

The seriesk=04k+132kcan be expressed ask=0449k.

The series k=0449kis in the standard form role="math" localid="1648976707514" k=0crkfor a geometric series with c=4and r=49.

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

Since r=49, it follows that the series role="math" localid="1648976736831" k=04k+132kconverges.

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 k=04k+132kconverges to 41-49, that is 365.

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