http://www.ck12.org Chapter 8. Infinite Series
Limit Comparison Test, Simplified Limit Comparison Test
Another test we can use to determine convergence of series without negative terms is theLimit Comparison Test.
It is easier to use than the Comparison Test.
Theorem (The Limit Comparison Test)
Suppose∑∞k= 1 ukis a series without negative terms. Then one of the following will hold.
- If∑∞k= 1 vkis a convergent series without negative terms and limk→∞vukkis finite, then∑∞k= 1 ukconverges.
- If∑∞k= 1 wkis a divergent series without negative terms and limk→∞vwkkis positive, then∑∞k= 1 ukdiverges.
The Limit Comparison Test says to make a ratio of the terms of two series and compute the limit. This test is most
useful for series with rational expressions.
Example 8Determine if∑∞k= 1 k^47 +k (^56) +k^3 K− 21 converges or diverges.
Solution
Just as with rational functions, the behavior of the series∑∞k= 1 k^47 +k (^56) +k^3 K− 21 whenkgoes to infinity behaves like the series
with only the highest powers ofkin the numerator and denominator:∑∞k= 17 kk^45. We will use the series∑∞k= 17 kk^45 to
apply the Limit Comparison Test. First, when we simplify the series∑∞k= 17 kk^45 , we get the series∑∞k= (^171) k. This is
a harmonic series because∑∞k= (^171) k=^17 ∑∞k= (^11) kand the multiplier^17 does not affect the convergence or divergence.
Thus,∑∞k= (^171) kdiverges. So, we will next check that the limit of the ratio of the terms of the two series is positive:
klim→∞
k^47 +k (^56) +k^3 k− 21
71 k =klim→∞
7 k^4 + 42 k^3 − 7
7 k^4 +K =^1 >^0.
Using the Limit Comparison Test, because 71 kdiverges and the limit of the ratio is positive, then∑∞k= 1 k^47 +k (^56) +k^3 K− 21
diverges.
Unlike the Comparison Test, you do not have to compare the terms of both series. You may just make a ratio of the
terms.
There is aSimplified Limit Comparison Test, which may be easier for you to use.
Theorem (The Simplified Limit Comparison Test)
Suppose∑∞k= 1 ukand∑∞k= 1 vkare series without negative terms. If limk→∞vukkis finite and positive, then either∑∞k= 1 uk
and∑∞k= 1 vkboth converge or∑∞k= 1 ukand∑∞k= 1 vkboth diverge.
Example 9Determine if∑∞k= 18 k^2 + 5 converges or diverges.
Solution
∑∞k= 18 k^2 + 5 is a series without negative terms. To apply the Simplified Limit Comparison Test, we can compare
∑∞k= 18 k^2 + 5 with the series∑∞k= 182 k, which is a convergent geometric series. Then limk→∞
8 k^2 + 5
82 k =limk→∞^8 k^8 +k^5 =^1 >0.
Thus, since∑∞k= 182 kconverges, then∑∞k= 18 k^2 + 5 also converges.
Review Questions
- Write an example of a nondecreasing sequence.