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Math Help - Intergral Test for Divergence

  1. #1
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    Intergral Test for Divergence

    1 + 1/6 + 1/11 + 1/16 +1/21

    The answer key tells me that the pattern in the denominator is 5x-4. Can someone explain how to find the pattern in this series?
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  2. #2
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    Quote Originally Posted by aaronb View Post
    1 + 1/6 + 1/11 + 1/16 +1/21

    The answer key tells me that the pattern in the denominator is 5x-4. Can someone explain how to find the pattern in this series?
    Look at the denominators 1, 6, 11, 16 and 21. The common jump is

     <br />
6=1 - 5, 11-6 = 5, 16-11 = 5<br />
which suggest a formula like a_n = 5n + b . Then subs n = 1 into a_n so a_1 = 1. This give b = -4. Then check to see if you get all the numbers a_1=1, a_2 =6,a_3=16 etc.
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  3. #3
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    Hello, aaronb!

    1 + \tfrac{1}{6} + \tfrac{1}{11} + \tfrac{1}{16} + \tfrac{1}{21} + \hdots

    The answer key tells me that the pattern in the denominator is 5n-4.
    Can someone explain how to find the pattern in this series?
    Obviously, we examine the denominators: . 1,\:6,\:11,\:16,\:21,\:\hdots

    Equally obvious is the fact that they "go up by 5."

    This is an Arithmetic Sequence with first term a = 1 and common difference d = 5.

    You're expected to the general term of an A.S.: . a_n \:=\:a + (n-1)d


    Hence, we have: . a_n \:=\:1 + (n-1)5 \quad\Rightarrow\quad\boxed{ a_n \:=\:5n-4}

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