Series converges or diverges

Question

∞
∑  ((3n-2)/(5n+1))^n
n=1

Answer 1

By the Root Test, ∑_{{from n=1 to n=∞}}[(3n − 2)/(5n + 1)]ⁿ will converge if the Limit

as n goes to positive infinity of {[(3n − 2)/(5n + 1)]ⁿ}^1/n is less than 1; the given series

will diverge if this same Limit exceeds 1.

Rewrite lim_(n→+∞){[(3n − 2)/(5n + 1)]ⁿ}^1/n as lim_(n→+∞)[(3n − 2)/(5n + 1)]¹ or

lim_(n→+∞)[(3 − 2/n)/(5 + 1/n)] or 3/5 equal to 0.6, which is less than 1.

A programmable calculator will run the given summation to 0.5128623492 in

48 "loops" of the program.