1. 11.4 The Comparison Tests In this section, we will learn:
INFINITE SEQUENCES AND SERIES
11.4
The Comparison Tests
In this section, we will learn:
How to find the value of a series
by comparing it with a known series.

2. COMPARISON TESTS
In the comparison tests, the idea is to compare a given series with one that is known to be convergent or divergent.

3. Consider the series This reminds us of the series .
COMPARISON TESTS
Series 1
Consider the series
This reminds us of the series
The latter is a geometric series with a = ½ and r = ½ and is therefore convergent.

4. COMPARISON TESTS
As the series is similar to a convergent series, we have the feeling that it too must be convergent.
Indeed, it is.

5. COMPARISON TESTS
The inequality
shows that our given series has smaller terms than those of the geometric series.
Hence, all its partial sums are also smaller than 1 (the sum of the geometric series).

6. COMPARISON TESTS
Thus,
Its partial sums form a bounded increasing sequence, which is convergent.
It also follows that the sum of the series is less than the sum of the geometric series:

7. COMPARISON TESTS
Similar reasoning can be used to prove the following test, which applies only to series whose terms are positive.
The first part says that, if we have a series whose terms are smaller than those of a known convergent series, then our series is also convergent.

8. COMPARISON TESTS
The second part says that, if we start with a series whose terms are larger than those of a known divergent series, then it too is divergent.

9. Suppose that  an and  bn are series with positive terms.
THE COMPARISON TEST
Suppose that  an and  bn are series with positive terms.
If  bn is convergent and an  bn for all n, then  an is also convergent.
If  bn is divergent and an  bn for all n, then  an is also divergent.

10. THE COMPARISON TEST - PROOF
Part i
Let
Since both series have positive terms, the sequences {sn} and {tn} are increasing (sn+1 = sn + an+1  sn).
Also, tn → t; so tn  t for all n.

11. THE COMPARISON TEST - PROOF
Part i
Since ai  bi, we have sn  tn. Hence, sn  t for all n.
This means that {sn} is increasing and bounded above.
So, it converges by the Monotonic Sequence Theorem.
Thus,  an converges.

12. THE COMPARISON TEST - PROOF
Part ii
If  bn is divergent, then tn → , since {tn} is increasing.
However, ai  bi; so sn  tn.
Thus, sn → ; so  an diverges.

13. SEQUENCE VS. SERIES
It is important to keep in mind the distinction between a sequence and a series.
A sequence is a list of numbers.
A series is a sum.

14. With every series  an, there are associated two sequences:
SEQUENCE VS. SERIES
With every series  an, there are associated two sequences:
The sequence {an} of terms
The sequence {sn} of partial sums

15. COMPARISON TEST
In using the Comparison Test, we must, of course, have some known series  bn for the purpose of comparison.

16. Most of the time, we use one of these:
COMPARISON TEST
Most of the time, we use one of these:
A p-series [ 1/np converges if p > 1 and diverges if p  1]
A geometric series [ arn–1 converges if |r| < 1 and diverges if |r|  1]

17. Determine whether the given series converges or diverges:
COMPARISON TEST
Example 1
Determine whether the given series converges or diverges:

18. For large n, the dominant term in the denominator is 2n2.
COMPARISON TEST
Example 1
For large n, the dominant term in the denominator is 2n2.
So, we compare the given series with the series
 5/(2n2).

19. since the left side has a bigger denominator.
COMPARISON TEST
Example 1
Observe that
since the left side has a bigger denominator.
In the notation of the Comparison Test, an is the left side and bn is the right side.

20. is convergent because it is a constant times a
COMPARISON TEST
Example 1
We know that
is convergent because it is a constant times a
p-series with p = 2 > 1.

21. Therefore, is convergent by part i of the Comparison Test.
Example 1
Therefore, is convergent by part i of the Comparison Test.

22. NOTE 1
Although the condition an  bn or an  bn in the Comparison Test is given for all n, we need verify only that it holds for n  N, where N is some fixed integer.
This is because the convergence of a series is not affected by a finite number of terms.
This is illustrated in the next example.

23. Test the given series for convergence or divergence:
COMPARISON TEST
Example 2
Test the given series for convergence or divergence:

24. COMPARISON TEST
Example 2
This series was tested (using the Integral Test) in Example 4 in Section 11.3
However, it is also possible to test it by comparing it
with the harmonic series.

25. Observe that ln n > 1 for n ≥ 3. So,
COMPARISON TEST
Example 2
Observe that ln n > 1 for n ≥ 3.
So,
We know that  1/n is divergent (p-series with p = 1).
Thus, the series is divergent by the Comparison Test.

26. NOTE 2
The terms of the series being tested must be smaller than those of a convergent series or larger than those of a divergent series.
If the terms are larger than the terms of a convergent series or smaller than those of a divergent series, the Comparison Test does not apply.

27. For instance, consider The inequality is useless as far as the
NOTE 2
For instance, consider
The inequality is useless as far as the
Comparison Test is concerned.
This is because  bn =  (½)n is convergent and an > bn.

28. NOTE 2
Nonetheless, we have the feeling that 1/(2n -1) ought to be convergent because it is very similar to the convergent geometric series  (½)n.
In such cases, the following test can be used.

29. Suppose that  an and  bn are series with positive terms. If
LIMIT COMPARISON TEST
Suppose that  an and  bn are series with positive terms. If
where c is a finite number and c > 0, either both series converge or both diverge.

30. LIMIT COMPARISON TEST - PROOF
Let m and M be positive numbers such that
m < c < M.
Since an/bn is close to c for large n, there is an integer N such that

31. LIMIT COMPARISON TEST - PROOF
If  bn converges, so does  Mbn.
Thus,  an converges by part i of the Comparison Test.
If  bn diverges, so does  mbn.
Thus,  an diverges by part ii of the Comparison Test.

32. Test the given series for convergence or divergence:
COMPARISON TESTS
Example 3
Test the given series for convergence or divergence:

33. We use the Limit Comparison Test with:
COMPARISON TESTS
Example 3
We use the Limit Comparison Test with:

34. COMPARISON TESTS
Example 3
We obtain:

35. This limit exists and  1/2n is a convergent geometric series.
COMPARISON TESTS
Example 3
This limit exists and  1/2n is a convergent geometric series.
Thus, the given series converges by the Limit Comparison Test.

36. Determine whether the given series converges or diverges:
COMPARISON TESTS
Example 4
Determine whether the given series converges or diverges:

37. The dominant part of the numerator is 2n2.
COMPARISON TESTS
Example 4
The dominant part of the numerator is 2n2.
The dominant part of the denominator is n5/2.
This suggests taking:

38. COMPARISON TESTS
Example 4
We obtain:

39.  bn = 2  1/n1/2 is divergent, p-series with p = ½ <1.
COMPARISON TESTS
Example 4
 bn = 2  1/n1/2 is divergent, p-series with p = ½ <1.
Thus, the given series diverges by the Limit Comparison Test.

40. COMPARISON TESTS
Notice that, in testing many series, we find a suitable comparison series  bn by keeping only the highest powers in the numerator and denominator.

41. ESTIMATING SUMS
We have used the Comparison Test to show that a series  an converges by comparison with a series  bn.
It follows that we may be able to estimate the sum  an by comparing remainders.

42. As in Section 11.3, we consider the remainder
ESTIMATING SUMS
As in Section 11.3, we consider the remainder
Rn = s – sn = an+1 + an+2 + …
For the comparison series  bn, we consider the corresponding remainder Tn = t - tn = bn+1 + bn+2 + …

43. As an  bn for all n, we have Rn  Tn.
ESTIMATING SUMS
As an  bn for all n, we have Rn  Tn.
If  bn is a p-series, we can estimate its remainder Tn as in Section 11.3
If  bn is a geometric series, then Tn is the sum of a geometric series and we can sum it exactly.
In either case, we know that Rn is smaller than Tn

44. Estimate the error involved in this approximation.
ESTIMATING SUMS
Example 5
Use the sum of the first 100 terms to approximate the sum of the series  1/(n3+1).
Estimate the error involved in this approximation.

45. the given series is convergent by the Comparison Test.
ESTIMATING SUMS
Example 5
Since
the given series is convergent by the Comparison Test.

46. ESTIMATING SUMS
Example 5
The remainder Tn for the comparison series  1/n3 was estimated in Example 5 in Section 11.3 (using the Remainder Estimate for the Integral Test).
We found that:

47. ESTIMATING SUMS
Example 5
Therefore, the remainder for the given series satisfies: Rn  Tn  1/2n2

48. ESTIMATING SUMS
Example 5
With n = 100, we have:
With a programmable calculator or a computer, we find that with error less than