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Integral Test

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Integral Test
Consider the series ∑n=N∞an\sum\limits^\infty_{n=N}a_nn=N∑∞​an​, where ana_nan​ is a sequence of positive terms. 
Let f(n)=anf(n)=a_nf(n)=an​ and assume that fff is continuous, positive and decreasing. 
Then

1. ∫N∞f(x)  ⁣dx{\displaystyle\int}^\infty_Nf(x)\de{x}∫N∞​f(x) dx converges if and only if ∑n=N∞an\sum\limits^\infty_{n=N}a_nn=N∑∞​an​ converges.

2. ∫N∞f(x)  ⁣dx{\displaystyle\int}^\infty_Nf(x)\de{x}∫N∞​f(x) dx diverges if and only if ∑n=N∞an\sum\limits^\infty_{n=N}a_nn=N∑∞​an​ diverges.

Identifying Clues
The terms ana_nan​ look integrable

Determine whether ∑n=1∞3nen2\sum\limits^\infty_{n=1}\frac{3n}{e^{n^2}}n=1∑∞​en23n​ converges or diverges.

It converges

It diverges

Cannot be determined

I don't know

Practice Question
Determine whether ∑n=3∞ln⁡nn\sum\limits^\infty_{n=3}\frac{\ln n}{n}n=3∑∞​nlnn​ converges or diverges.

It converges

It diverges

Cannot be determined

I don't know