I don't understand this explanation for #\sum_(n=0)^\infty((-1)^n)/(5n-1)#? Why test for convergence/divergence AGAIN, if the Limit Comparison Test confirms that both series are the same?

"First we will check whether the series is absolutely convergent.
Because If the series is absolutely convergent then the series is convergent.
But if the series is not absolutely convergent. Then we will check whether the series is convergent or divergent."




Answer 1

The series only converges conditionally. It is possible to have #sum|a_n|# diverge and #suma_n# converge; it is called conditional convergence.

The Limit Comparison Test, when applied to this problem, tells us that the absolute value of the series, #sum_(n=0)^oo|a_n|,# diverges.
However, it is totally possible to have #sum_(n=0)^oo|a_n|# diverge and #sum_(n=0)^ooa_n# converge.

This is called conditional convergence , and we must always check for it.

There is no "absolute divergence" that tells us if the absolute value of the series diverges, so must the series itself.

So, if #a_n=(-1)^n/(5n+1), |a_n|=1/(5n+1)# as the absolute value omits the alternating negative signs.
Now, we can say #b_n=1/(5n)#, and we know #sum_(n=0)^oo1/(5n)=1/5sum_(n=0)^oo1/n# diverges, it is a #p#-series with #p=1#.

Now, using the Limit Comparison Test,

#c=lim_(n->oo)|a_n|/b_n=(1/(5n+1))/(1/(5n))=lim_(n->oo)(5n)/(5n+1)=1>0neoo#

So, both series must diverge.

Now, the original series, #sum_(n=0)^oo(-1)^n/(5n+1)# is an alternating series -- we are not taking the absolute value; we are acknowledging that it has negative terms.
Now, the positive, non-alternating portion of #a_n=(-1)^n/(5n+1)# is #b_n=1/(5n+1)#

The Alternating Series Test tells us if

#lim_(n->oo)b_n=0, b_n>=b_(n+1)# (IE #b_n# is decreasing), then the series converges.
#1/(5n-1)# decreases as #n# grows due to the growing denominator; #lim_(n->oo)1/(5n-1)=0#, so the series converges.
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Answer 2

Testing for convergence/divergence again, even after confirming that two series are the same using the Limit Comparison Test, is done to ensure the validity of the comparison. While the Limit Comparison Test helps establish that the given series behaves similarly to a known series, it does not definitively prove convergence or divergence on its own. By applying another convergence test, such as the Alternating Series Test or the Ratio Test, we can independently verify the convergence or divergence of the series under consideration. This additional step adds rigor to the analysis and ensures the accuracy of the conclusions drawn regarding the convergence or divergence of the series.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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