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Chapter 12: Q 59. (page 825)

In Problems 51鈥�66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

鈭�k=1鈭�514k-1

Short Answer

Expert verified

The given infinite geometric series 鈭�k=1鈭�514k-1is convergent and its sum is 203.

Step by step solution

01

Step 1. Write the given information.

The given geometric series is:

鈭�k=1鈭�514k-1

02

Determine the first term and common ratio.

The first term isa1=5

The common ratio is the ratio of successive terms:

r=14

03

Step 3. Determine whether the infinite geometric series is converges or diverges.

If r<1then the infinite geometric series 鈭�k=1鈭�a1rk-1converges.

As we can see that r=14and14<1, therefore the given series is convergent.

04

Step 4. Find the sum of the series.

Use the formula to find the sum of the given geometric series鈭�k=1鈭�a1rk-1=a11-r,

鈭�k=1鈭�514k-1=51-14=534=203

Therefore, the sum of the infinite geometric series is203.

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