calculus – Van Wijngaarden’s trick positive series in alternate serie

Designed for series in which every term is positive, it uses van Wijngaarden’s trick for converting the series into an alternating one the following
$$sum _{k=0}^{infty } a(k)=sum _{n=0}^{infty } (-1)^n sum _{k=0}^{infty } 2^{k-1} aleft(k 2^{k-1}right)$$ it possible to prove it in analitical way, the proof I found is his use in an algoritm