The pinch theorem is used to find the limit, Xn=(A1^n+A2^n+…… +AK ^ n) to the power of N, where a1 > A2 > >Ak>0

The pinch theorem is used to find the limit, Xn=(A1^n+A2^n+…… +AK ^ n) to the power of N, where a1 > A2 > >Ak>0

Let y = lnxn
Then y = 1 / N * ln [(A1) ^ n + (A2) ^ n + + (An)^n]
= 1/n * ln{(A1)^n * [1 + (A2/A1)^n + (A3/A1)^n + …… + (An/A1)^n]
= 1/n * ln(A1)^n + 1/n * ln[1 + (A2/A1)^n + (A3/A1)^n + …… + (An/A1)^n]
= lnA1 + 1/n * ln[1 + (A2/A1)^n + (A3/A1)^n + …… + (An/A1)^n]
So, limy
=lim lnA1 + lim 1/n * ln[1 + (A2/A1)^n + (A3/A1)^n + …… + (An/A1)^n]
=lnA1 + lim 1/n * ln[1 + 0 + 0 + …… +Note: 1 > a > 0, when n →∞, Lima ^ n = 0
=lnA1
So,
limXn = lim e^y
=e^(limy)
=e^(lnA1)
=A1