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:
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