Find whether the series converges or diverges using tests

[lucid acorn]

Question in latex

[young grove]

1. Do not ping the Moderators, unless someone is breaking the rules.
2. Do not ping the Helper Moderators, unless there is a conflict between helpers.
3. Do not ping other members randomly for help.
4. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
5. Wait patiently for a helper to come along.
6. If the Helper has answered your question, remember to thank them with the Mathematics Ranks bot and close the thread with:

+close
Feel free to nominate the person for helper of the week in #helper-nominations
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[lucid acorn]

$$ \sum_{n=2}^{\infty} (\frac{n-1}{n+2})^{n(n-1)}$$

[urban vapor]

Yooda

[lucid acorn]

I've used both the ratio and root test and got 1 for both the results

and ive tried the integral test but i dont see a way that i can integrate this simply

[civic tusk]

What about the test for divergence?

You have a 1^infinity indeterminate form as the limit of the summand.

[lucid acorn]

these are the only 3 tests ive been taught

usually for the first too L would either be greater or less than 1 to find convergence or divergence but it was 1 both times so its neither

[civic tusk]

You were taught the integral test before the test for divergence?

[lucid acorn]

yes unless im missing it in my notes and stuff

its just these 3

[civic tusk]

...okay, weird.

[lucid acorn]

this one?

[civic tusk]

Yes.

[lucid acorn]

unless i missed it in one of my lectures then i never did but it seems simple enough

therefore the series diverges since it equates to 1^\infty right?

[oblique fjord]

No, that’s just an indeterminate form.

An indeterminate form can converge to 0.

So you should evaluate the limit with some other techniques.

[dry hollow]

Hint : $\underset{n \to \infty}{lim} (\frac{n-1}{n+2})^{n-1}=e^{-3}$ try bounding the term in the series in some way, try comparing to the geometric series

[urban vapor]

Rotor 🙂

[lucid acorn]

+close

[heavy dirge]

Please thank your Helpers before closing!

Please thank the helpers who assisted you by clicking the buttons below. You can thank each helper only once. Once you're done, click "Close Post" to close this thread.

[heavy dirge]

Thank you for your feedback! Kocher has been awarded 1 . They now have 1022 . They have 1 daily left for today.

[heavy dirge]

Thank you for your feedback! Techie Literate has been awarded 1 . They now have 911 . They have 1 daily left for today.