Product Notation Problem

Question

Can Someone help me with these on product Notation?

The notation $ \prod _ { i = 1 } ^ N p _ i $ denotes the product with $ N $ factors: $$ \prod _ { i = 1 } ^ N p _ i = p _ 1 p _ 2 \dots p _ N \text . $$ Compute the following products.

1. $ \prod _ { i = 1 } ^ M \frac 1 \theta $
2. $ \prod _ { k = 1 } ^ K \frac k { k + 1 } $
3. $ \ln \left( \prod _ { k = 1 } ^ K e ^ k \right) $

Answer 1

I'll just give you hints. The first is a product of $M$ apparently equal factors. The second is a telescoping product. For the third, use $\ln\prod_ke^{f(k)}=\sum_k\ln e^{f(k)}=\sum_kf(k)$.