Define and explain Reciprocal Series

Question

What is meant by a Reciprocal Series? In my textbook it's been defined as "A series whose every term is composed of the reciprocal of the product of r factors in Arithmetic series, the first factors of all denominators being in the same Arithmetic series." I get it to some extent but still confused. Please explain.

Answer 1

What is meant by a Reciprocal Series? In my textbook it's been defined as...

Didn't see it under that name specifically, but here is how I would parse the given definition.

"A series whose every term is composed of the reciprocal of the product of r factors in Arithmetic series,

$r$ factors in arithmetic progression can be written as $a(a+b)(a+2b)\ldots(a+(r-1)b)$ for some $a$ (first term) and $b$ (common difference).

The reciprocal of such a product would be $\dfrac{1}{a\big(a+b\big)\big(a+2b\big)\ldots\big(a+(r-1)b\big)}\,$.

the first factors of all denominators being in the same Arithmetic series."

If the $k^{th}$ term is $\dfrac{1}{a_k(a_k+b)(a_k+2b)\ldots(a_k+(r-1)b)}\,$, then the first factor of the denominator is $a_k\,$, and these first factors are said to be in "the same" arithmetic progression, so $a_k = a_0+kb\,$.

Then the $k^{th}$ term is $\dfrac{1}{\big(a_0+kb\big)\big(a_0+(k+1)b\big)\big(a_0+(k+2)b\big)\ldots\big(a_0+(k+r-1)b\big)}\,$, and the series would be:

$$ \begin{align} \sum_{k \ge 0}\dfrac{1}{\big(a_0+kb\big)\big(a_0+(k+1)b\big)\big(a_0+(k+2)b\big)\ldots\big(a_0+(k+r-1)b\big)} = \sum_{k \ge 0} \prod_{j=0}^{r-1} \dfrac{1}{a_0+(k+j)b} \end{align} $$