limn→∞(n+1n)(−1)n=．

Question

lim

n→∞

((n+1)/(n))(−1)n=______．

Answer 1

∵((n+1)/(n))(−1)n＝e(−1)nln(n+1)/(n)
∴

lim

n→∞

((n+1)/(n))(−1)n＝

lim

n→∞

eln((n+1)/(n))(−1)n＝e

lim

n→∞

(−1)nln((n+1)/(n))，
而数列{（-1）ⁿ}有界，

lim

n→∞

ln((n+1)/(n))＝0，
∴

lim

n→∞

(−1)nln((n+1)/(n))＝0．
故 

lim

n→∞

((n+1)/(n))(−1)n＝e0＝1．