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Chapter 7: Q 53. (page 615)

Determine whether the series 鈭�k=2鈭�-35kconverges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series 鈭�k=2鈭�-35k converges to 940.

Step by step solution

01

Step 1. Given information.

Given a series 鈭�k=2鈭�-35k.

02

Step 2. Find if the series converges or not.

The index starts with 2, 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 鈭�k=0鈭�crk.

Here, the series role="math" localid="1648964110412" 鈭�k=2鈭�-35khas c=925and r=-35.

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

Since r=-35, it follows that the series role="math" localid="1648964322077" 鈭�k=2鈭�-35kconverges.

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=2鈭�-35kconverges to 9251--35, that is 940.

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