Question

Use the integral test to determine whether the series En=1- infinity of 1/(3n +1)

Sorry I don't have the latex script to properly type this on my iPad. Please let me know if the question doesn't make sense

Answer 1

U will need to find this integral:
\[\int\limits_{1}^{\infty} \frac{ 1 }{ 3n+1 }\]
If this integral converges, your series converges too. The only criteria is that the function is positive valued, continuous and decreasing. Check, check, check

Answer 2

That integral is ,
1/3 ln (3n+1) +C

Answer 3

As
\[\lim_{n \rightarrow \infty} \ln n =\infty \]
thus your series is divergent.

Answer 4

Thank you, Andrus!!

Answer 5

No problem

Answer 6

*Andras

Answer 7

The answer is correct, just as a side note:
Make sure that in an exam you first verify that the sequence \(a_n\geq 0, \ \forall n \in \mathbb{N}\) and that the series is monotone decreasing \(a_{n+1} \leq a_n, \ \forall n \in \mathbb{N}\) otherwise you can't apply the integral test.

Answer 8

@Andras has already mentioned that in his post, just to give those lines credits here, try \(f'(x) \leq 0\) by transforming the sequence into a real valued function \(\mathbb{R} \to \mathbb{R}\)

Answer 9

I did :) It was the check, check, check part

Answer 10

Thank you both!!