Divergence Test

#Calc-2#Math#Series If a series converges, then the ${\lim }_{k\to \infty }{S}_k=L$ ${\lim }_{k\to \infty }{S}_{k-1}=L$ ${\lim }_{k\to \infty }(S_k-S_{k-1})={\lim }_{k\to \infty }{a}_k=0$ So if the series converges, then the final limit is true. If the final limit does not equal 0, then the series diverges.

This is a one way statement, if $\sum _{k=1}^{\infty }{a}_k$ converges THEN the limit = 0. BUT if the limit = 0, there’s still a chance the series could diverge A better way of using this test is if the ${\lim }_{k\to \infty }{a}_k\ne 0$, then the series diverges. If it equals 0, the series might converge or diverge.

Examples §