Question

How do I know when to use limit comparison test and direct comparison test?

Answer 1

HINT: These tests are helpful to find the convergent and divergent series and we have to use these tests in different conditions and for that some criteria is to be followed and every series are not solved by every method we have to do some calculations to justify which method is appropriate for solving the required series.

Complete answer:
We will the study the two tests in detail so that we get the knowledge of choosing the which test is more convenient for solving the series.
According to direct Comparison test,
We have to suppose that we have two series \[\sum {{a_n}} \] and \[\sum {{b_n}} \] with \[{a_n},{b_n} \geqslant 0\] for all \[n\] and \[{a_n} \leqslant {b_n}\]for all \[n\] then,
1. If \[\sum {{b_n}} \] is convergent then \[\sum {{a_n}} \] also convergent
2. If \[\sum {{a_n}} \] is divergent then \[\sum {{b_n}} \] also divergent
In simple words, this test is nothing but, if we have $2$ series of positive terms & the terms of one of the series is always greater than the terms of the other series also, if the greater series is convergent the smaller series must converge also, if the smaller series is divergent then the larger series must diverge.
According to limit comparison test,
We have to suppose that we have two series \[\sum {{a_n}} \] and \[\sum {{b_n}} \] with \[{a_n} \geqslant 0\& {b_n} > 0\] for all \[n\] then
value of is taken as \[c = \mathop {\lim }\limits_{n \to \infty } \dfrac{{{a_n}}}{{{b_n}}}\] .
1. If value of \[c\] is positive and finite then either both the series converge or both the series diverge.
2. If \[c = 0\] and \[\sum {{b_n}} \] converges then \[\sum {{a_n}} \] also converges.
3. If \[c = \infty \] and\[\sum {{b_n}} \] diverges then \[\sum {{a_n}} \] also diverges.
In this test of limit comparison we do not worry about which terms of the series comes in numerator and which comes in denominator because we get same result either you put any terms in numerator and any term in denominator.
The pro of the limit comparison test is, we can compare the series without verifying the inequality we need and to apply the direct comparison test, we have to evaluate the given limit.

Note:
A series is said to be convergent if the ${n^{th}}$ term converges to $0$ and a series is divergent if it is not convergent, a series is said to be divergent if the ${n^{th}}$ term converges to $0$ and a series is convergent if it is not divergent. A series is convergent (or converges) only if the sequence of its partial sums tends to a limit.