The series 1-1+0+1-1+0+1-1+0+1-1+0+ ... (Grandi's Series) by inserting 0s into this sum....

Question:

The series 1-1+0+1-1+0+1-1+0+1-1+0+ ... (Grandi's Series) by inserting 0s into this sum. Likewise, if we were to add zero's to a convergent series:

a) Would it still remain convergent?

b) Would the sum of the series change?

Infinite Series and Partial sums:

This problem involves the understanding of infinite series. Here, zeros have been inserted in the Grandi's series. The thing to understand is that Grandi's series is not convergent and thus adding zeros will not make it one. Secondly, the sum is simply the Cesaro sum of the Grandi series and the value certainly changes due to the insertion of zeros.

Answer and Explanation:

(a) Note, the Grandi's series is a divergent series. It's Cesaro's sum is equal to 0.5.

(b) Yes, adding zeros certainly changes the Cesaro sum of the series. Because you simply can't use the shifting and adding method now. The idea is to check through following the steps in the method of calculating the Cesaro's sum. The basic thing to understand is, if the series is non-convergent but yet converges to a sum, then insertion of zeros can change the value.

Note, the new sum tends to go to (1/3)


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Infinite Series & Partial Sums: Explanation, Examples & Types

from GRE Math: Study Guide & Test Prep

Chapter 12 / Lesson 4
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