Summation Series to Product Series Part 1.

Here's something interesting I found out last year. I don't know if its been done before but I strongly suspect it has.

Consider a summation series of form $S(n) = \sum_{k = 1}^{k = n}\frac{f(k)}{k!}$ such that $f(1) = 1$.

Now let's do something unusual and create the following sequence starting with the term $a_2 = 2$ and then $a_{k+1} = (k + 1)(a_k + f(k))$ for all $k > 2$.

Finally we'll focus our attention on the following product series $P(n) = \prod_{k=2}^{k=n} ( 1 + \frac{f(k)}{a_k})$ for all $n > 2$.

We'll show that $S(n) = P(n)$ for all $n > 2$.

Proof : By induction on n. We'll start with the base case. Base Case: $n = 2$

We have $P(2) = 1 + \frac{f(2)}{a_2} = 1 + \frac{f(2)}{2} =\frac{f(1)}{1} + \frac{f(2)}{2} = S(2)$ as required.

Induction Step

This part is a quite long, so we'll break it down into 3 parts.

Part 1 : Re-arranging the Factors of the Product Series

We can write each factor from our product series as $\begin{aligned} 1 + \frac{f(k)}{a_k} &= \frac{a_k}{a_k} + \frac{f(k)}{a_k} \\&= \frac{a_k + f(k)}{a_k} \\&= \frac{a_{k+1}}{(k+1)a_k} \end{aligned}$ by using the definition of our sequence.

Part 2 : Re-writing our Product Series

Now we have that $\begin{aligned} \prod_{k = 2}^{k = n} (1 + \frac{f(k)}{a_k}) &= \frac{a_{3}}{(3)a_2} \times \frac{a_{4}}{(4)a_3} \times ...\times \frac{a_{n+1}}{(n+1)a_n} \\ &= \frac{a_{n+1}}{a_2 \times 3 \times 4 \times ... \times n \times (n+1)} \\&=\frac{a_{n+1}}{(n+1)!} \end{aligned}$

Part 3 : Final Step

Suppose $P(q) = S(q)$ for some natural number $q$.

Then$\begin{aligned} P(q+1) &= P(q)(1 + \frac{f(q+1)}{a_{q+1}}) \\&= P(q) + \frac{f(q+1)}{a_{q+1}}\times\frac{a_{q+1}}{(q+1)!} \\&= P(q) + \frac{f(q+1)}{(q+1)!} \\&= T(q) + \frac{f(q+1)}{(q+1)!} = T(q+1) \end{aligned}$ as required.

End of Proof

We'll go through some of the consequences of this in the next note!