help for root test for series

[mint topaz]

i dont understand how
|a_n| <= p^(n-N)|a_N|
is obtained. Could someone please enlighten me?

[nimble marsh]

(an/an-1)(an-1/an-2)....product of N terms

[gritty wolf]

let p = (1 + r)/2

what you want to consider is that |a_(n + 1)|/|a_n| =< (1 + r)/2 = p

but there is also something that may lead to the inequality you want

that is you can pick an N so large than |a_n|/|a_N| =< p

or this is equivalent to |a_n| =< p|a_N|

now from the below inequality we get |a_n| =< p^(n - N)|a_N| which seems to be missing

but you can observe that this also works for |a_(n + 1)|/|a_n|

i am sure some sort of proof by induction will suffice here