Examples on Identifying Geometric Series and Determining if it Conv...

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In this video, I go over two examples and two methods of identifying if a series is a geometric series, and then finding its sum if it converges. In the first example, we are given the first 4 terms of a series, which we can identify the first term and calculate its common ratio from each successive term. Since the absolute value of the common ratio is less than 1, we can obtain its sum. I also illustrate in a graph how the partial sums of this series oscillate back and forth while ultimately approaching the limit of 3, which is defined as its actual sum.

In the second example, I re-arrange the summation equation directly to show that it is a geometric series. In this case, the common ratio is greater than 1, so the series diverges.

#math #calculus #series #geometricseries #education

3 Geometric Series Examples.png

Timestamps

Example 2: Sum of geometric series when given first 4 terms – 0:00
- Solution: First term a is 5 and common ratio r is -2/3 – 0:20
- |r| is less than 1 so the series is a convergent geometric series and its sum is a/(1-r) = 3 – 1:34
- Sum of an infinite series is the limit the sum approaches as the number of terms approaches infinity – 3:20
- Table and graph of the partial sums as they oscillate around and approach the sum of 3 – 3:45
Example 3: Determining if a series is convergent or divergent – 5:06
- Solution: Re-arrange the summation to get it in the familiar form for a geometric series – 5:33
- Series is a geometric series with initial term 4 and common ratio 4/3, which is greater than 1, so the series diverges – 8:36
- Alternatively, we could identify the series by writing out the first few terms – 9:00
- a = 4 and r = 4/3 – 11:05