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Question
Show that `lim_("n" -> oo) (1^2 + 2^2 + ... + (3"n")^2)/((1 + 2 + ... + 5"n")(2"n" + 3)) = 9/25`
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Solution
`lim_("n" -> oo) (1^2 + 2^2 + ... + (3"n")^2)/((1 + 2 + ... + 5"n")(2"n" + 3)) = lim_("n" -> oo) ((3"n"(3"n" + 1)(2 xx 3"n" + 1))/6)/((5"n"(5"n" + 1))/2 (2"n" + 3))`
= `lim_("n" -> oo) (3"n"(3"n" + 1)(6"n" + 1) xx 2)/(6 xx 5"n"(5"n" + 1)(2"n" + 3))`
= `lim_("n" -> oo) ("n"(3"n" + 1)(6"n" + 1))/(5"n"(5"n" + 1)(2"n" + 3))`
= `lim_("n" -> oo) ("n"*"n"(3 + 1/"n") "n"(6 + 1/"n"))/(5"n"*"n"(5 + 1/"n") "n"*(2 + 3/"n"))`
= `lim_("n" -> oo) ("n"^3(3 + 1/"n")(6 + 1/"n"))/("n"^3*5(5 + 1/"n")(2 + 3/"n"))`
= `((3 + 0)(6 + 0))/(5(5 + 0)(2 + 0))`
= `18/50`
= `9/25`
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