Question
Prove that the sum of all natural numbers is -1/12.
Asked by: USER3535
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Answer (51)
This is not true. The infinite series
[tex]\displaystyle\sum_{n=1}^\infty n[/tex]
converges if and only if the sequence of its partial sums converges. The [tex]k[/tex]-th partial sum is
[tex]\displaystyle\sum_{n=1}^kn=\frac{k(k+1)}2[/tex]
but clearly this diverges as [tex]k[/tex] gets arbitrarily large.
[tex]\displaystyle\sum_{n=1}^\infty n[/tex]
converges if and only if the sequence of its partial sums converges. The [tex]k[/tex]-th partial sum is
[tex]\displaystyle\sum_{n=1}^kn=\frac{k(k+1)}2[/tex]
but clearly this diverges as [tex]k[/tex] gets arbitrarily large.