Question

The product of (1-1/n)(1-1/n+1)(1-1/n+2)....(1-1/2n) is equal to​

Answer 1

Step-by-step explanation:

We have

pk=∏n=2k(1−1n2)=∏n=2k(n−1)(n+1)n2=1⋅32⋅2⋅2⋅43⋅3⋅3⋅54⋅4⋅⋯⋅(k−2)⋅k(k−1)⋅(k−1)⋅(k−1)(k+1)k⋅k=12(1+1k)

because all but the first and last numerators and denominators cancel. Therefore

∏n=2∞(1−1n2)=limk→∞pk=12