Answer:
The series converges, and the limit of bn when n→∞ is 0
Step-by-step explanation:
- A) The nth term of this series is [tex] b_n=\frac{(-1)^n}{n5^n} [/tex]. We use the ratio test to decide the convergence of the series. Let [tex] L=\lim_{n\rightarrow \infty}\frac{|b_n|}{|b_{n+1}|} [/tex]. Then [tex]L=\lim_{n\rightarrow \infty}\frac{n5^n}{(n+1)5^{n+1}}=\lim_{n\rightarrow \infty}\frac{1}{5}\frac{n}{n+1}=\frac{1}{5} [/tex]. We obtain that [tex]L<1[/tex] therefore the series converges.
- B) If we had [tex] \lim_{n\rightarrow \infty} b_n \neq 0 [/tex] then by the divergent series test, the series [tex] \sum b_n [/tex] would be divergent which contradicts part A). We conclude that this limit must be 0.
