Why the series is divergent based on the Preliminary test

In summary, the conversation discusses the use of preliminary test to determine the divergence of a series. It is noted that if the limit is not equal to zero, the series is divergent, but if it is equal to zero, further testing is required. The conversation also mentions the use of the ##(-1)^n## factor in the test and the oscillating behavior of series number 3, which requires further testing. Finally, the person asks for help in applying the test and later realizes their mistake in stating that series number 3 tends to both 1 and -1. The homework question involves using the preliminary test to determine the divergence of three series, including series number 3 which requires further testing.
  • #1
agnimusayoti
240
23
Homework Statement
Use the preliminary test to decide whether the following series are divergent or require further testing:
1. ##\sum_1^\infty {\frac {(-1)^{n+1}n^2}{n^2+1}}##
2. ##\sum_1^\infty {\frac {(-1)^n n^2}{(n+1)^2}}##
3. ##\sum_1^\infty {\frac {(-1)^n n}{\sqrt{n^3+1}}}##
Relevant Equations
Preliminary test:
if ##\lim_{n\to \infty} a_n \neq 0## or ##\lim_{n\to \infty} a_n## does not exist then the series is divergent. If ##\lim_{n\to \infty} a_n = 0## then the series need further testing.
Interestingly, If I neglect the ##(-1)^n## or ##(-1)^{n+1}## then apply preliminary test, I could find the limit. Whether the limit is not equal to zero, as in series number 1 and 2, then I can conclude the series is divergent. But, if the limit is equal to zero, as in series number 3, then I can conclude the series need further testing.

I try to use the correct preliminary test, that including the ##(-1)^n## factor, then I try to find the limit as n approaching infinity. If I expand those 3 series, the sign is oscillating. Maybe, I can conclude that the limit of the series doesn't exist (or the sequences do not tend to some finite number). So the series is divergent according to preliminary test. But, surprisingly, series number 3 is oscillating too, and tends to -1 or 1 (depends on odd/even terms); yet it need further testing (which fit with explanation in the 1st paragraph).

At this point, I just realize that I don't have sufficient skill to apply preliminary test in the alternating series. Could you please guys, help me? Thanks a lot.
 
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  • #2
Guys, I think I have the answer from another thread. I'm very glad. Thanksss.
 
  • #3
Look again at whether series 3 tends to 1 or -1 as you say.

Ah, if I am not mistaken you also say two opposite things in the same post, first you say it tends to 0, later that it alternates between 1 and -1.
agnimusayoti said:
Homework Statement:: Use the preliminary test to decide whether the following series are divergent or require further testing:
...
3. ##\sum_1^\infty {\frac {(-1)^n n}{\sqrt{n^3+1}}}##
Relevant Equations:: , if the limit is equal to zero, as in series number 3, then I can conclude the series need further testing.

surprisingly, series number 3 is oscillating too, and tends to -1 or 1 (depends on odd/even terms); yet it need further testing (which fit with explanation in the 1st paragraph).
 
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Related to Why the series is divergent based on the Preliminary test

1. Why is the series divergent based on the Preliminary test?

The series is divergent based on the Preliminary test because it fails to meet the necessary conditions for convergence. The Preliminary test states that for a series to converge, the terms of the series must approach zero as n approaches infinity. If the terms do not approach zero, then the series will not converge and is therefore divergent.

2. What is the Preliminary test and how does it determine convergence or divergence?

The Preliminary test, also known as the Divergence test, is a method used to determine the convergence or divergence of a series. It states that if the limit of the terms of a series does not approach zero as n approaches infinity, then the series will diverge.

3. Can a series be divergent based on the Preliminary test even if the terms approach zero?

Yes, a series can still be divergent based on the Preliminary test even if the terms approach zero. This is because the test only determines if the series is convergent or divergent, not if it approaches a specific value. If the terms approach zero too slowly, the series may still diverge.

4. Are there any exceptions to the Preliminary test?

Yes, there are some exceptions to the Preliminary test. For example, the test is not applicable to alternating series or series with terms that alternate between positive and negative values. In these cases, other tests, such as the Alternating Series test, must be used to determine convergence or divergence.

5. Can a series be convergent even if it fails the Preliminary test?

Yes, a series can still be convergent even if it fails the Preliminary test. This is because the test only determines if the series is divergent, not if it is convergent. Other tests, such as the Ratio test or the Integral test, can be used to determine convergence even if the Preliminary test fails.

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