True-False Quiz Question 22: Multiplication of Terms of Series May NOT Equal Multiplication of Sums

In this video I show that if we have two infinite series whose sums are A and B, then a third series comprising of both terms multiplied by each other may not sum to AB. I show this by considering two convergent Geometric Series. The first series has terms (0.1)^n and the sum is 1/9. The second series has terms (0.2)^n and the sum is 1/4. Thus AB = 1/36. If we multiplied the terms of each series to form a new series we get a series with terms (0.02)^n. The sum of this new series, using the formula for the sum of a convergent Geometric Series, is 1/49. But this is not the same as the value AB = 1/36. Thus the answer to our question is False.

Time stamps:

• Question 22: 0:00
• Solution: False: 0:38
• Testing question with Geometric Series: 0:53
  • First Series: Sum = A = 1/9: 1:25
  • Second Series: Sum = B = 1/4: 2:56
  • Sum of multiplication series = 1/49: 4:03
  • Multiplying sums = AB = 1/36: 6:48

Full video below:

• Infinite Sequences and Series: Review and True-False Quiz:
• HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
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