Does the series # sum_{n=1}^oo (5n)^(3n)/(5^n+3)^n# converge or diverge?

1 Answer
Mar 18, 2018

Answer:

See below.

Explanation:

Considering

#a_n = (5n)^(3n)/(5^n+3)^n#

we have

#a_n < (5n)^(3n)/(5^n)^n#

now we will compare #a_n# with #1/n^2# that we know to converge.

Taking #ln#

#ln a_n < (3n)ln(5+ln n)-n^2 ln 5# and as we can easily verify exists a #n_0# such that #n > n_0 rArr ln a_n < ln (1/n^2)#

As we know, #ln# is a strictly increasing function so considering #n > n_0#

#lna_n < ln (1/n^2) rArr a_n < 1/n^2#

hence the series

# sum_{n=1}^oo (5n)^(3n)/(5^n+3)^n#

converges.