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Chapter 7: Q 47. (page 615)

Determine whether the series 鈭�k=0鈭�32kconverges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series鈭�k=0鈭�32kconverges to 6.

Step by step solution

01

Step 1. Given information.

Given a series鈭�k=0鈭�32k.

02

Step 2. Find if the series converges or not.

The series 鈭�k=0鈭�32kis in the standard form localid="1648980136761" role="math" 鈭�k=0鈭�crkfor a geometric series with localid="1648886392804" c=3and r=12.

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

Since localid="1648886396247" r=12, it follows that the series鈭�k=0鈭�32kconverges.

03

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

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

So, the series鈭�k=0鈭�32kconverges to31-12, that is 6.

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