What is the limit of the sequence

Question

limit n->infinity −(7.2^n+5.5^n/7+3.5^(n+1))?

Answer 1

The intuition is that we divide out one 3.5 from the denominator, then we have something larger in the numerator to the n-th power over something smaller in the denominator to the same power, hence something > 1 to the n-th power, and therefore that the limit is -∞ because of the minus sign in front.

 

First, deal with the 7 in the denominator.

 

-(7.2ⁿ+5.5ⁿ)/[7 + 3.5ⁿ⁺¹] < -(7.2ⁿ+5.5ⁿ)/[3.5ⁿ⁺²]   for n >= 0, because 3.5ⁿ⁺² > 7+3.5ⁿ⁺¹ for n >= 0.

 

= -(1/(3.5)²)(7.2ⁿ+5.5ⁿ)/[3.5ⁿ]

< -(1/16)(7.2/3.5)ⁿ

< -(1/16)2ⁿ

= -2^n-4

 

and lim_n→∞ -2^n-4 = lim_m→∞ -2^m = -∞.

 

Since the latter limit diverges to minus infinity, so does the initial limit