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Which of the following statements is false?

A

If $\lim _{ n\rightarrow \infty }{ { a }_{ n } } =0$, then $\sum _{ n=1 }^{ \infty }{ { a }_{ n } } $ converges.

B

If $\sum _{ n=1 }^{ \infty }{ { a }_{ n } } =L$, then $\sum _{ n=0 }^{ \infty }{ { a }_{ n } } =L+{ a }_{ 0 }$.

C

If $\left| r \right| <1$, then $\sum _{ n=0 }^{ \infty }{ a{ r }^{ n } } =\frac { a }{ 1-r } $.

D

The series $\sum _{ n=1 }^{ \infty }{ \cfrac { n }{ 100(n+2) } } $ diverges.

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