Hello, I have 3 questions here, hopefully someone can please help me to check on their convergence and limit.
1. sequence = (ln n / ln 2n)
the limit I calculated is 1. therefore, convergent.
2. sequence = (1+[2/n]) ^ (1/n)
the limit i calculated is 1. therefore, convergent.
3. sequence = (1+[3/n]) ^ n
the limit i calculated is 1. therefore, convergent.
Thank you very much in advance.
1. sequence = (ln n / ln 2n)
the limit I calculated is 1. therefore, convergent.
2. sequence = (1+[2/n]) ^ (1/n)
the limit i calculated is 1. therefore, convergent.
3. sequence = (1+[3/n]) ^ n
the limit i calculated is 1. therefore, convergent.
Thank you very much in advance.
-
1) lim(n→∞) ln n / ln(2n)
= lim(n→∞) (1/n) / (2/(2n)), by L'Hopital's Rule
= 1.
2) lim(n→∞) (1 + 2/n)^(1/n) = 1^0 = 1.
3) lim(n→∞) (1 + 3/n)^n
= lim(n→∞) [(1 + 3/n)^(n/3)]^3
= lim(t→∞) [(1 + 1/t)^t]^3, letting t = n/3
= e^3, using the limit definition for e.
I hope this helps!
= lim(n→∞) (1/n) / (2/(2n)), by L'Hopital's Rule
= 1.
2) lim(n→∞) (1 + 2/n)^(1/n) = 1^0 = 1.
3) lim(n→∞) (1 + 3/n)^n
= lim(n→∞) [(1 + 3/n)^(n/3)]^3
= lim(t→∞) [(1 + 1/t)^t]^3, letting t = n/3
= e^3, using the limit definition for e.
I hope this helps!