how do i prove this converges?

[glass moss]

i tried to prove tht sn of this term diverges but that didnt work out i had no luck in comparison tests either...

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[burnt pewter]

what's the general term?

[glass moss]

(n+1/n)^2-(n+1/n)

power -n

[burnt pewter]

$a_n =\left( \frac{(n+1)^{n+1}(n)-n^{n+1}(n+1)}{n^{n+1} (n)} \right) ^{-n}$

@glass moss

take the common terms out

[gaunt orbit]

CoffeyOnExile

[glass moss]

we take n out?

the limit comes out to be 0

so we cant tell if its convergent or not yet

[burnt pewter]

yes we can

[glass moss]

but we get limit as 0?

[burnt pewter]

that's what we want

[glass moss]

but our instructor said tht limit is =/ 0 then the series is divergent if limit is 0 then test is inconclusive

[burnt pewter]

did you understand what he meant?

or no

like the reasoning behind this

[glass moss]

we did a proof for this statement using the help of cauchy sequence rules too

i think i understood what he said

[burnt pewter]

$a_n = n \left( \frac{n^{2n}}{n(n+1)((n+1)^n - n^n)} \right) ^n$

[gaunt orbit]

CoffeyOnExile

[glass moss]

omfg i am doing a lot of silly mistakes