Calculus of a Single Variable, Chapter 9, 9.3, Section 9.3, Problem 1

sum_(n=1)^oo1/(n+3)
The integral test is applicable if f is positive, continuous and decreasing function on infinite interval [k,oo) where k>=1 and a_n=f(x) . Then the series sum_(n=1)^ooa_n converges or diverges if and only if the improper integral int_1^oof(x)dx converges or diverges.
For the given series a_n=1/(n+3)
Consider f(x)=1/(x+3)
Refer to the attached graph of the function. From the graph,we observe that the function is positive, continuous and decreasing for x>=1
Let's determine whether function is decreasing by finding the derivative f'(x),
f'(x)=-1(x+3)^(-2)
f'(x)=-1/(x+3)^2
f'(x)<0 which implies that the function is decreasing.
We can apply the integral test, since the function satisfies the conditions for the integral test.
Now let's determine whether the corresponding improper integral converges or diverges,
int_1^oo1/(x+3)dx=lim_(b->oo)int_1^b1/(x+3)dx
=lim_(b->oo)[ln|x+3|]_1^b
=lim_(b->oo)ln|b+3|-ln|1+3|
=lim_(b->oo)ln|b+3|-ln4
lim_(b->oo)(b+3)=oo
Apply the common limit:lim_(u->oo)(ln(u))=oo
=oo-ln4
=oo
Since the integral int_1^oo1/(x+3)dx diverges, we can conclude from the integral test that the series diverges.