The "shift & add" technique is an excellent trick to get standard formulas for summation of various finite sequences. e.g.
(Newbie here, couldn't get the aligning proper, so stuck to special case $n=5$) $$ \begin{aligned} 1-2+3-4+5&=S\\ 1-2+3-4&=S-5 \end{aligned} $$ And, we could add up & get the results/formula accordingly. ($\sum_{k=1}^\infty(-1)^{k-1}k=\frac{1}{2}$)
But, is this trick suitable or correct when dealing with infinite sequences? e.g. $$ \begin{aligned} 1-2+3-4+5-\cdots&=S\\ 1-2+3-4+\cdots&=S \end{aligned} $$ The professor then adds up!? (isn't the result off by infinity?!) If this is correct, the source link result is quite astounding indeed, what more mathematical implications does it have?
Source - ASTOUNDING: $1 + 2 + 3 + 4 + 5 + \cdots = -{\frac{1}{12}}$
