Let us start from your last equation, and focus specifically on the expectation. Assuming that the end date of each period is the start period of the next, the idea is to simplify it using conditional expectations.

Since $t < t_{n-2}$, we can write: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})} \times \frac{1}{P(t_{n-1}, t_{n})} \right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}\times \Bbb{E}_{t_{n-2}}^{Q^{t_n}} \left[ \frac{P(t_{n-1}, t_{n-1})}{P(t_{n-1}, t_{n})} \right]\right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}\times \frac{P(t_{n-2}, t_{n-1})}{P(t_{n-2}, t_{n})} \right]\\ \end{aligned} $$

We can see that the product is getting smaller, since the term $P(t_{n-2}, t_{n-1})$ that appeared in the numerator will simplify with the the last term of the product. $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-3} \frac{1}{P(t_i, t_{i+1})}\times \frac{1}{P(t_{n-2}, t_{n})} \right]\\ \end{aligned} $$ By repeating this operation, the product disappears (assuming that at pricing date $t$, the swap didn't start yet), and you get: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\frac{1}{P(t_{0}, t_{n})} \right]\\ &= \frac{P(t, t_0)}{P(t, t_n)}\\ \end{aligned} $$

This ratio can also be written as a product of capitalization factors using Libor forwards as follows: $$ \frac{P(t, t_0)}{P(t, t_n)} = \prod_{i=0}^{n-1} \frac{P(t, t_i)}{P(t, t_{i+1})} = \prod_{i=0}^{n-1} 1 + d(t_i, t_{i+1}) L(t, t_i, t_{i+1}) $$

Let us start from your last equation, and focus specifically on the expectation. Assuming that the end date of each period is the start period of the next, the idea is to simplify it using conditional expectations.

Since $t < t_{n-2}$, we can write using the tower property of conditional expectations: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})} \times \frac{1}{P(t_{n-1}, t_{n})} \right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[ \Bbb{E}_{t_{n-2}}^{Q^{t_n}} \left[ \underbrace{\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}}_{\mathcal{F}_{t_{n-2}}-\text{measurable}} \times \frac{1}{P(t_{n-1}, t_{n})} \right] \right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}\times \Bbb{E}_{t_{n-2}}^{Q^{t_n}} \left[ \underbrace{\frac{P(t_{n-1}, t_{n-1})}{P(t_{n-1}, t_{n})}}_{\mathbb{Q}^{t_n}\text{martingale}} \right]\right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}\times \frac{P(t_{n-2}, t_{n-1})}{P(t_{n-2}, t_{n})} \right]\\ \end{aligned} $$

We can see that the product is getting smaller, since the term $P(t_{n-2}, t_{n-1})$ that appeared in the numerator will simplify with the the last term of the product. $$ \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] = \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-3} \frac{1}{P(t_i, t_{i+1})}\times \frac{1}{P(t_{n-2}, t_{n})} \right] $$ By repeating this operation, the product disappears (assuming that at pricing date $t$, the swap didn't start yet, i.e.: $t < t_0$), and you get: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\frac{P(t_0, t_0)}{P(t_{0}, t_{n})} \right]\\ &= \frac{P(t, t_0)}{P(t, t_n)}\\ \end{aligned} $$

This ratio can also be written as a product of capitalization factors using Libor forwards as follows: $$ \frac{P(t, t_0)}{P(t, t_n)} = \prod_{i=0}^{n-1} \frac{P(t, t_i)}{P(t, t_{i+1})} = \prod_{i=0}^{n-1} 1 + d(t_i, t_{i+1}) L(t, t_i, t_{i+1}) $$

Let us start from your last equation, and focus specifically on the expectation. Assuming that the end date of each period is the start period of the next, the idea is to simplify it using conditional expectations.

Since $t < t_{n-2}$, we can write: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})} \times \frac{1}{P(t_{n-1}, t_{n})} \right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}\times \Bbb{E}_{t_{n-2}}^{Q^{t_n}} \left[ \frac{P(t_{n-1}, t_{n-1})}{P(t_{n-1}, t_{n})} \right]\right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}\times \frac{P(t_{n-2}, t_{n-1})}{P(t_{n-2}, t_{n})} \right]\\ \end{aligned} $$

We can see that the sum is getting smaller, since the term $P(t_{n-2}, t_{n-1})$ that appeared in the numerator will simplify with the the last term of the product. $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-3} \frac{1}{P(t_i, t_{i+1})}\times \frac{1}{P(t_{n-2}, t_{n})} \right]\\ \end{aligned} $$ By repeating this operation, the product disappears (assuming that at pricing date $t$, the swap didn't start yet), and you get: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\frac{1}{P(t_{0}, t_{n})} \right]\\ &= \frac{P(t, t_0)}{P(t, t_n)}\\ \end{aligned} $$

This ratio can also be written as a product of capitalization factors using Libor forwards as follows: $$ \frac{P(t, t_0)}{P(t, t_n)} = \prod_{i=0}^{n-1} \frac{P(t, t_i)}{P(t, t_{i+1})} = \prod_{i=0}^{n-1} 1 + d(t_i, t_{i+1}) L(t, t_i, t_{i+1}) $$

Let us start from your last equation, and focus specifically on the expectation. Assuming that the end date of each period is the start period of the next, the idea is to simplify it using conditional expectations.

Since $t < t_{n-2}$, we can write: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})} \times \frac{1}{P(t_{n-1}, t_{n})} \right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}\times \Bbb{E}_{t_{n-2}}^{Q^{t_n}} \left[ \frac{P(t_{n-1}, t_{n-1})}{P(t_{n-1}, t_{n})} \right]\right]\\ &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-2} \frac{1}{P(t_i, t_{i+1})}\times \frac{P(t_{n-2}, t_{n-1})}{P(t_{n-2}, t_{n})} \right]\\ \end{aligned} $$

We can see that the product is getting smaller, since the term $P(t_{n-2}, t_{n-1})$ that appeared in the numerator will simplify with the the last term of the product. $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-3} \frac{1}{P(t_i, t_{i+1})}\times \frac{1}{P(t_{n-2}, t_{n})} \right]\\ \end{aligned} $$ By repeating this operation, the product disappears (assuming that at pricing date $t$, the swap didn't start yet), and you get: $$ \begin{aligned} \Bbb{E}_{t}^{Q^{t_n}} \left[\prod_{i=0}^{n-1} \frac{1}{P(t_i, t_{i+1})} \right] &= \Bbb{E}_{t}^{Q^{t_n}} \left[\frac{1}{P(t_{0}, t_{n})} \right]\\ &= \frac{P(t, t_0)}{P(t, t_n)}\\ \end{aligned} $$

This ratio can also be written as a product of capitalization factors using Libor forwards as follows: $$ \frac{P(t, t_0)}{P(t, t_n)} = \prod_{i=0}^{n-1} \frac{P(t, t_i)}{P(t, t_{i+1})} = \prod_{i=0}^{n-1} 1 + d(t_i, t_{i+1}) L(t, t_i, t_{i+1}) $$