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The fact that series (infinite sums) such as

\displaystyle \sum_{n\geq1} 1 = 1+1+1+1+1+1+\cdots

\displaystyle \sum_{n\geq1} n = 1+2+3+4+5+6+\cdots

\displaystyle \sum_{n\geq0} 2^n = 1+2+4+8+16+32+\cdots

are diverging towards infinity (therefore have an infinite sum), and series such as

\displaystyle \sum_{n\geq1} (-1)^{n-1} = 1-1+1-1+1-1\pm\cdots

keep on oscillating between two or more values (therefore apparently do not have a sum), and series such as

\displaystyle \sum_{n\geq1} (-1)^{n-1} \, n = 1-2+3-4+5-6\pm\cdots

\displaystyle \sum_{n\geq0} (-1)^n \, 2^n = 1-2+4-8+16-32\pm\cdots

are diverging towards both +\infty and -\infty simultaneously (therefore series that apparently do not have a sum, or have (unsigned) infinity as a sum) is the common “wisdom”.

However, all divergent series (both those diverging towards +\infty or -\infty, and those whose partial sums keep on oscillating between values, and those diverging towards both +\infty and -\infty simultaneously) do have their own intrinsic finite sum, reflecting the fact that there is no infinity is nature. Here “intrinsic” means that such sum does not depend on the summation method or on any other calculation method used to determine such sum. It is truly the only finite sum consistent with the body of number theory (and of mathematics in general) that can be associated to the series.

Likewise, all infinite products do have their own intrinsic finite value.

This website presents the outcome of several years of research on divergent series and divergent infinite products, including hundreds of sums of divergent series and hundreds of values of infinite products. For example:

\displaystyle \sum_{n\geq1} 1 = 1+1+1+1+1+1+\cdots = -\dfrac{1}{2}

\displaystyle \sum_{n\geq1} n = 1+2+3+4+5+6+\cdots = -\dfrac{1}{12}

\displaystyle \sum_{n\geq0} 2^n = 1+2+4+8+16+32+\cdots = -1

\displaystyle \sum_{n\geq1} (-1)^{n-1} = 1-1+1-1+1-1\pm\cdots = \dfrac{1}{2}

\displaystyle \sum_{n\geq1} (-1)^{n-1} \, n = 1-2+3-4+5-6\pm\cdots = \dfrac{1}{4}

\displaystyle \sum_{n\geq0} (-1)^n \, 2^n = 1-2+4-8+16-32\pm\cdots = \dfrac{1}{3}

We also present a taxonomy of divergent series and infinite products whose purpose is to try organising these series into families, tribes, genera, sections, species and varieties, similar to the taxonomies used in life science (botany and zoology).