Theorem
29
[238] Let \(b_ n\) be a sequence for which
\[ b_{n}β₯ b_{n+1} {\gt}0\quad , \quad \lim _{nββ } b_{n} = 0 \quad , \]
then the series
\[ β _{n=0}^{+β }{ (-1)^{n}b_{n}} \]
is convergent; also, called \(β\) the value of the series, letting
\[ B_ N = β _{n=0}^{N }{ (-1)^{n}b_{n}} \]
the partial sums, we have that the sequence \(B_{2N}\) is decreasing , the sequence \(B_{2N+1}\) is increasing, and both converge to \(β\).