Series are a common question type on math tests. Series questions can be either algebra or geometry based, but most often appear as algebra questions.
Algebra series questions typically ask you to determine the value of a particular term in the series or determine if the series is convergent or divergent. Convergence or divergence is based on whether the terms in the series get closer or farther apart as more terms are added.
For example, if 1 + 1 + 1 + 1 + 1 + . . . gets farther and farther apart as more numbers are added, then the series is convergent. If the numbers get closer and closer together, then the series is divergent.
This question will focus on how to determine whether a given series is convergent or divergent.
Test for convergence
When dealing with infinite series, convergence is just as important as absolute value when dealing with finite sums.
Convergence is determined by the value of the series term by term. As mentioned before, if the series term decreases continually, the series is decreasing and thus convergent.
Convergence can also be determined by looking at the ratio of two consecutive terms. If this ratio is close to one, then the series is convergent. For example, if 1 + 1 2 = 1 + 0.5 = 1 / 1.5 = 0.8 , then the series is convergent because the ratio of terms is close to one (0.8 ≈ 1).
Another way to determine convergence is to see if the sequence terminates and then check if the remaining numbers in order are all equal to one another.
Series illustration
A good way to visualize a series is to imagine it as a sequence of terms, or values, that add up to a final value. For example, the infinite series
1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + ⋯
can be thought of as the following sequence of terms:
1, 2, 3, 4, 5, 6, 7, 8, 9,…
In this case the last term (9) adds up to the infinite value 8. This can be visualized as placing 9 boxes next to each other—they all add up to the same length as the infinite series above.
Convergent series
A series that eventually settles into a single value is called a convergent series. There are two main tests for convergence: the ratio test and the integral test.
The ratio test says that the series is convergent if the ratios of any two subsequent terms always converge to some value. For example, if the difference between each term is constant, then that constant difference will always be less than or equal to one, so the series will converge.
The integral test says that the series is convergent if its sum can be integrated (replaced with an integral symbol). Ultimately, this test says that if you can add up all of the values in the sequence, then it must be a specific value. If this is not true, then the series is not convergent.
These are just two of many tests for convergence! Try experimenting with some other types of series and seeing if you can determine whether they are convergent or divergent.
Divergent series
A series that does not converge is called a divergent series. This means that the values obtained by summing the infinite number of terms in the series do not necessarily have a value.
For example, the infinite series 1 + 1 2 2 + 1 3 3 + 1 4 4 + 1 5 5 ⋯ is divergent, since it can be proved by algebra that its sum is infinity.
The question of whether a given series is convergent or divergent depends only on the pattern of its terms, not on their values.
For instance, the infinite series 1 − 1 2 −1 3−1 4−1 5−1 6−1 ⋯ is also divergent, since it can be proved by algebra that its sum is minus one. {|class=”wikitable” border=”0″ cellpadding=”5″ style=”font-size: 11px; line-height: 9px; margin: 0; padding: 0; border: 0; font-family: inherit; vertical-align: top;” !More Info |- !Series!What to look for!
Series with no constant difference
Series with no constant term
Parallel sequences
Arithmetic sequences with common difference = 0
Examples
- Analyze each type of sequence individually to determine convergence or divergence.
- If x n = a n /b and b ≠ 0 then x n converges if lim n → ∞ a n /b = 0 and x n diverges if limn→∞a n /b >0.
- If {an}n=1 then {an}n converges if an=0 and diverges if an≠0.
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