Examples

Simple divergence test: examples

(a) Consider: $\sum _{n=1}^{\infty }\frac{n}{4n+1}$

This diverges by the SDT because $a_n\to \frac{1}{4}$ and not $0$.

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(b) Consider: $\sum _{n=1}^{\infty }(-1{)}^{n-1}\frac{n}{n+1}$

This diverges by the SDT because ${\lim }_{n\to \infty }{a}_n=\text{DNE}$.

We can say the terms “converge to the pattern $+1,-1,+1,-1,\dots$,” but that is not a limit value.