>>12639366Ramanujan summation is a way to quantify the strength of a divergence of a series. For example,
1-1+1-1+1-1+1-1+... = y
1-y = y
y = 1/2
1+2+4+8+...=y
1+2y=y
y = -1
Interestingly, Ramanujan summation provides a valuable tool for gauging whether a series is convergent. Theorem:
If a series' Ramaujan sum, R, has a value of R <= 1, the series is divergent.
Corollary:
If a series' Ramanujan summation is negative, R < 0, then we say the original series is divergent toward either +infinity or -infinity.
Corollary:
When R < 0, then |R| quantifies the speed at which the series diverges toward +- infinity.
Lemma:
If a series is Ramanujan convergent, then its Ramanujan sum, R, is equivalent to its sum.
