How do you use the integral test to determine if 1/3+1/5+1/7+1/9+1/11+... is convergent or divergent?
1 Answer
First, we have to write a rule for this summation. Note the denominator is increasing by
sum_(n=0)^oo1/(2n+3)
There are other summations that would work here as well, but this will suffice.
In order for
f(n) must be positivef(n) must be decreasing
Both of these are true: all the terms are greater than
The integral test states that for
So, we will evaluate the following integral:
int_0^oo1/(2x+3)dx=lim_(brarroo)int_0^b1/(2x+3)dx
color(white)(int_0^oo1/(2x+3)dx)=1/2lim_(brarroo)int_0^b2/(2x+3)dx
color(white)(int_0^oo1/(2x+3)dx)=1/2lim_(brarroo)[ln(|2x+3|)]_0^b
color(white)(int_0^oo1/(2x+3)dx)=1/2lim_(brarroo)(ln(2b+3)-ln(3))
As
color(white)(int_0^oo1/(2x+3)dx)=oo
The integral diverges, so