1. the nsider, for n → 1, the sequence an given by an = n log (n/n+1) Determine the limit of the sequence as n→∞, If it exists , or explain why the sequence diverges. In your answers include the names of any rules, theorems or limits you have used. 2. Relevant equations 3. The attempt at a solution lim n→∞ an = lim n→∞ n log(n/n+1) embed sequence in function f(x)= x log (x/(x+1)) now limx→∞ x log(x/x+1) which is (∞ . ∞) form, can rearange to give = limx→∞ log(x/(x+1)) / (1/x) which gives ( 0 / 0 ) indeterminate form, can then use log laws on the top function to rewrite log(x/(x+1)) as log(x)-log(x+1) = limx→∞ log(x) - log (x+1) / (1/x) now differntiate numerator and denominator by l'hopital rule. = limx→∞ ((1/x) -(1/(1+x)) / (-1 /x^2) The x(1+x) terms cancel out when you flip and multiply. Leaving = limx→∞ 1 x (-1) = -1 Therefore the limit approaches -1 and converges to -1. Using theorem limx→∞ f(x) = L => lim n→∞ an = L. So limn→∞ an converges to -1. Any help or tips would be appreciated as my Maths is really bad.