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Math Lesson 12.4.2 - Converging and Diverging Infinite Series
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Welcome to our Math lesson on Converging and Diverging Infinite Series, this is the second lesson of our suite of math lessons covering the topic of Infinite Series Explained, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Converging and Diverging Infinite Series Explained
Some number series have a finite value despite having an infinite number of terms. They are called converging infinite series. For example, geometric series with a fractional common ratio smaller than 1 are infinite series, as the terms become smaller and smaller; the increase in sum becomes more and more irrelevant. A typical infinite series is
This series points towards the value 1, as first we take half of the whole, then half of the remaining half and so on, as illustrated in the figure below.
This is because the divisions remain always within the original square, regardless of the number of partitions made. Hence, we can write
The condition for an infinite series to be convergent is that
where L is a finite number. (In the previous example L = 1.)
Example 1
What is the value of
Solution 1
From the example, in theory, we can write
= (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ⋯)-1/2
= 1 - 1/2
= 2/2 - 1/2
= 1/2
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