# Thread: Intergral Test for Divergence

1. ## 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?

2. Originally Posted by aaronb
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

$
6=1 - 5, 11-6 = 5, 16-11 = 5
$
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.

3. 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: . $a_n \:=\:1 + (n-1)5 \quad\Rightarrow\quad\boxed{ a_n \:=\:5n-4}$