How do you use the integral test to determine if #1/(sqrt1(sqrt1+1))+1/(sqrt2(sqrt2+1))+1/(sqrt3(sqrt3+1))+...1/(sqrtn(sqrtn+1))+...# is convergent or divergent?
1 Answer
The series is divergent. See the explanation below.
Explanation:
This is the series:
#sum_(n=1)^oo1/(sqrtn(sqrtn+1))#
Before starting with the integral test, we need to see that a couple conditions are met first.
In order for
#f(n)# must be decreasing on#n in [N,oo)# #f(n)>0# for#n in [N,oo)#
For
The integral states that if the improper integral
So, in order to test the convergence of the given series, we need to evaluate the integral
#int1/(sqrtx(sqrtx+1))dx" "" "# Let:#{(u=sqrtx+1),(du=1/(2sqrtx)dx):}#
#=2int1/(2sqrtx(sqrtx+1))dx=2int1/udu=2lnabsu+C#
Then:
#int_1^oo1/(sqrtx(sqrtx+1))dx" "" "" "# (Take the limit at infinity.)
#=lim_(brarroo)int_1^b1/(sqrtx(sqrtx+1))dx=lim_(brarroo)(2lnabsu)|_1^b#
#=lim_(brarroo)2lnb-2ln1#
As
#=oo#
The integral diverges. Through the integral test, the series
Note that
This can be alternatively be compared via the limit comparison test with
This is just as valid a method as the integral test, but quicker.