Comparison+Test

When do you use a comparison test?
You should use a comparison test when the function you are testing is similar to another function which you have already tested for convergence or divergence. Similar functions are any which behave in similar manners, such as those with the same power. For example, 1/n 3 is similar to the function 3/(n3 +2).

Direct Comparison Test
The direct comparison test is used if the function you are testing can be proven to be always less than or equal to OR always greater than or equal to a known converging or diverging function. However, if a function is sometimes greater than and sometimes less than the known function, then you must use the limit comparison test. For example, the function 1/n3  is always less than 1/( n3  +4), so the direct comparison test can be applied. Conversely, 1/n3  is less than 1/( <span style="background: white; color: black; font-family: Arial; font-size: 16px; text-align: left; text-decoration: none;">n<span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 12px; text-align: left; text-decoration: none; vertical-align: super;">3  <span style="background-color: transparent; color: black; font-family: Arial; font-size: 16px; text-align: left; text-decoration: none;">-4) for the value n=1 and greater than 1/( <span style="background: white; color: black; font-family: Arial; font-size: 16px; text-align: left; text-decoration: none;">n<span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 12px; text-align: left; text-decoration: none; vertical-align: super;">3  <span style="background-color: transparent; color: black; font-family: Arial; font-size: 16px; text-align: left; text-decoration: none;">-4) for the value n=2. In this second case, the direct comparison test can NOT be used, but instead the limit comparison test can.

Limit Comparison Test
<span style="background-color: transparent; color: black; display: block; font-family: Arial; font-size: 16px; text-align: left; text-decoration: none;">The limit comparison test can be used when the direct comparison test is invalid. In this test, the function to be tested is referred to as a<span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 10px; text-align: left; text-decoration: none; vertical-align: sub;">n and the known function is referred to as b n. You should take the limit of a<span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 10px; text-align: left; text-decoration: none; vertical-align: sub;">n / b<span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 10px; text-align: left; text-decoration: none; vertical-align: sub;">n as n approaches infinity. If this is evaluated to a positive, finite number, than the two limits have the same behavior (either converging or diverging). If the limit is not equal to a positive, finite number, then the opposite is true and the function tested has the opposite behavior as the known function. Note: The **harmonic series** is the divergent infinite series used in the above example and shown below. [])

More Problems Determine whether each series converges or diverges. 1) ................................ 2) ......................... 3) ....................... 4) ............................... 5) .........

Solutions [|Problem_1.jpeg] [|Problem_2.jpeg] [|Problem_3.jpeg] [|Problem_4.jpeg] [|Problem_5.jpeg]

Sources: <span style="color: #000000; font-family: 'Times New Roman',Times,serif; font-size: 110%;">__ [] __ <span style="background-color: transparent; color: #810081; display: block; font-family: 'Times New Roman',Times,serif; font-size: 16px; text-align: left;">__[]__ <span style="background-color: transparent; color: #810081; display: block; font-family: 'Times New Roman',Times,serif; font-size: 16px; text-align: left;">[]

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