An interesting fact is that the Root Test (stated at the end of the previous problem) is stronger than the Ratio Test. Proving this fact would require showing two things: First, one can show that any time the Ratio Test limit exists and equals some L then the Root Test limit also exists and equals the same L. Second, one can find examples of series where the Root Test works but the Ratio Test does not. Your task in this problem is to do the second thing above. To be more specific, and an example of a seriesnan such that

• limnnanexists and equals a number less than 1, meaning the series converges absolutely by the Root Test, and
• limnnanan+1does not exist.

Give a brief explanation why your example satisfies the above two properties.