# How do you use the integral test to determine if #ln2/2+ln3/3+ln4/4+ln5/5+ln6/6+...# is convergent or divergent?

##### 1 Answer

#### Answer

#### Answer:

#### Explanation

#### Explanation:

#### Answer:

It is divergent. See explanation.

#### Explanation:

First write the series:

#ln2/2+ln3/3+ln4/4+...=sum_(n=2)^ooln(n)/n#

Before getting into the integral test, we must assure two things first: for the integral test to apply to

Both of these are true since

Furthermore,

So, we see the integral test applies. The integral test states that if the two aforementioned conditions are met, then for

If the integral converges to a real, finite value, then the series converges. If the integral diverges, then the series does too.

So, we take the integral

#int_2^ooln(x)/xdx=lim_(brarroo)int_2^bln(x)/xdx#

Letting

#=lim_(brarroo)int_ln(2)^ln(b)ucolor(white).du#

#=lim_(brarroo)[1/2u^2]_ln(2)^ln(b)#

#=lim_(brarroo)1/2ln^2(b)-1/2ln^2(2)#

As

#=oo#

The integral diverges. Thus, we see that

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