Suppose that (an) satisfies an #0 Vn N and lim =r ...

 

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Suppose that (an) satisfies an #0 Vn N and lim =r <1. an Prove that Clan converges and in particular an converges. (a) Show that there exists kEN such that for all nk we have lan+1 Sans where s= (b) Lets and k be as above. Show that for all nk we have landslags"-k (c) Show that s <1 and use this to prove that the tail al converges using the comparison test. Explain why this implies that the series) la converges. (d) Use the Ratio Test to show that the series n=1 IMB EIN IM converges for all r R. Suppose that (an) satisfies an #0 Vn N and lim =r <1. an Prove that Clan converges and in particular an converges. (a) Show that there exists kEN such that for all nk we have lan+1 Sans where s= (b) Lets and k be as above. Show that for all nk we have landslags"-k (c) Show that s <1 and use this to prove that the tail al converges using the comparison test. Explain why this implies that the series) la converges. (d) Use the Ratio Test to show that the series n=1 IMB EIN IM converges for all r R.