od Daniel » Sreda, 14. Februar 2018, 02:48
U opštem slučaju, sume [inlmath]\sum\limits_{k=0}^\infty A(k)[/inlmath] i [inlmath]\sum\limits_{k=1}^\infty A(k)[/inlmath] (gde je [inlmath]A(k)[/inlmath] bilo koji izraz u kojem figuriše [inlmath]k[/inlmath]) naravno da nisu jednake. Razlikuju se za onaj sabirak u kojem je [inlmath]k=0[/inlmath], tj.
[dispmath]\sum_{k=0}^\infty A(k)=A(0)+\underbrace{A(1)+A(2)+A(3)+\cdots}_{\sum\limits_{k=1}^\infty A(k)}=A(0)+\sum_{k=1}^\infty A(k)[/dispmath] Međutim, u ovoj konkretnoj sumi je nulti sabirak [inlmath]A(0)[/inlmath] jednak nuli:
[dispmath]A(0)=(-1)^0\cdot0=0[/dispmath] zbog čega su i sume [inlmath]\sum\limits_{k=0}^\infty(-1)^k\cdot k[/inlmath] i [inlmath]\sum\limits_{k=1}^\infty(-1)^k\cdot k[/inlmath] jednake.
Možemo to zapisati i ovako:
[dispmath]\sum_{k=1}^\infty(-1)^k\cdot k=(-1)^1\cdot1+(-1)^2\cdot2+(-1)^3\cdot3+\cdots\\
\sum_{k=0}^\infty(-1)^k\cdot k=\underbrace{(-1)^0\cdot0}_0+\underbrace{(-1)^1\cdot1+(-1)^2\cdot2+(-1)^3\cdot3+\cdots}_{\sum\limits_{k=1}^\infty(-1)^k\cdot k}=\sum_{k=1}^\infty(-1)^k\cdot k[/dispmath]
I do not fear death. I had been dead for billions and billions of years before I was born, and had not suffered the slightest inconvenience from it. – Mark Twain