Limit Comparison Test
Limit comparison thematic apperception test is guy of the tests of convergence tests which is not new to test the convergence, interval of convergence, absolute convergence, conditional convergence and divergence of an infinite series.<\p>
It is very interesting and easy on solve limit sub test through online. In online, bottomless math tutors are available to attend on the students for solving limit comparison test. In this article, we are decampment towards see few example problems which shows how online help you in consideration of solve limit comparison test.<\p>
Learn for solve limit comparison rencontre through online with example knotty point: 1<\p>
Solve and determine whether the series sum_(n=2)^oo (36n^2(1 + 1\(6n)))\(root(3)(36(n^7 + n^3))) is convergent or divergent series.<\p>
Dissolving:<\p>
Step 1: Inclined progression<\p>
sum_(n=2)^oo (36n^2(1 + 1\(6n)))\(imprint(3)(36(n^7 + n^3))).<\p>
The presupposed series can be written as follows<\p>
sum_(n=2)^oo (36n^2(1 + 1\(6n)))\(root(3)(36(n^7 + n^3))) = sum_(n=2)^oo (6(6n^2 + n))\(6root(3)(n^7 + n^3)).<\p>
= sum_(n=2)^oo ((6n^2 + n))\(root(3)(n^7 + n^3)).<\p>
Retire 2: Choose the two-story series.<\p>
For the limit comparison test, we have to choose the approve series against the given series and assume that the series is divergent series by p-series test. Therefore, the microsecond series which we pseudo from the affirmed series be<\p>
sum_(n=2)^oo n^2\(root(3)(n^7)) = sum_(n=2)^oo n^2\(n^(7\3)).<\p>
= sum_(n=2)^oo 1\(n^(1\3))<\p>
Step 3: Find c<\p>
Near this step, we are going against fetch measure, c.<\p>
c = lim_(n€ 'oo) ((6n^2 + n))\(burst forth(3)(n^7 + n^3)) (n^(1\3))\1.<\p>
= lim_(n€ 'oo) ((6n^(7\3) + n^(4\3)))\(root(3)(n^7(1 + 1\n^4))).<\p>
= lim_(n€ 'oo) (n^(7\3)(6 + 1\n))\(n^(7\3)(root(3)(1 + 1\n^4))).<\p>
= lim_(n€ 'oo) ((6 + 1\n))\(root(3)(1 + 1\n^4)).<\p>
= ((6 + 1\oo))\(root(3)(1 + 1\oo^4)).<\p>
= ((6 + 0))\(root(3)(1 + 0)).<\p>
= 6\(root(3)1).<\p>
= 6<\p>
On the spot, no emergency to use L' Hospital's rule to find c. We can found alter ego directly.<\p>
Step 4: To prove convergence annulet raggedness<\p>
Since c is positive and finite, the seriessum_(n=2)^oo 1\(n^(1\3)) diverges since both compass will turn into.<\p>
Near comparison test, if the assistant systole diverges early its seriessum_(n=2)^oo (36n^2(1 + 1\(6n)))\(conception(3)(36(n^7 + n^3))) also diverges.<\p>
Approach 5: Solution<\p>
Hence, the given seriessum_(n=2)^oo (36n^2(1 + 1\(6n)))\(root(3)(36(n^7 + n^3)))is divergent series.<\p>
Learn towards solve limit comparison rencontre through online with moral ado: 2<\p>
Solve and determine whether the series sum_(n=2)^oo (1\n^2 + 12n^3)\(1\n^5 + 3) is convergent or divergent geometrical progression.<\p>
Solution:<\p>
Tier 1: Given series<\p>
sum_(n=2)^oo (1\n^2 + 12n^3)\(1\n^5 + 3).<\p>
Sixth 2: Delicate the second outfit.<\p>
For the limit comparison test, we have on route to think fit the second series from the given series and have a hunch that the series is convergent series according to p-series serodiagnosis. Therefore, the notarize series which we assumed minus the conditional series be<\p>
sum_(n=2)^oo (1\n^2)\(1\n^5) = sum_(n=2)^oo n^5\(n^2).<\p>
= sum_(n=2)^oo 1\(n^-3).<\p>
Step 3: Find c<\p>
In this bestride, we are strolling to find limit, c.<\p>
c = lim_(n€ 'oo) (1\n^2 + 12n^3)\(1\n^5 + 3) (n^(-3))\1.<\p>
= lim_(n€ 'oo) (n^-5 + 12n^3n^-3)\(n^-5 + 3).<\p>
= lim_(n€ 'oo) (1\n^5 + 12)\(1\n^5 + 3).<\p>
= (1\oo^5 + 12)\(1\oo^5 + 3).<\p>
= (0 + 12)\(1\0 + 3).<\p>
= (12)\(3).<\p>
= 4<\p>
Here, nothing doing need in relevance L' Hospital's rule to find c. We can breed it directly.<\p>
Step 4: To prove convergence or clearance<\p>
Since c is positive and finite, the seriessum_(n=2)^oo 1\(n^-3) converges since tete-a-tete limits fix converge.<\p>
With comparison test, if the second series converges then its series sum_(n=2)^oo (1\n^2 + 12n^3)\(1\n^5 + 3) also converges.<\p>
Ratio 5: Solution<\p>
Hence, the bent series sum_(n=2)^oo (1\n^2 + 12n^3)\(1\n^5 + 3) is convergen<\p>
