Final Answer:
Without specific values for the common ratio or the first term in the infinite geometric series, it's impossible to determine its convergence or calculate the sum. Therefore, the answer is 4) Divergent. Thus the correct option is 4) Divergent.
Step-by-step explanation:
An infinite geometric series converges if its common ratio (r) is between -1 and 1; otherwise, it diverges. For a series with a constant sum, the sum (S) is calculated as
, where 'a' is the first term and 'r' is the common ratio.
None of the given options provide information about the common ratio (r) or the first term (a). Hence, without specific values for 'r' or 'a', it's impossible to accurately determine the convergence or the sum of the series. Therefore, the answer is 4) Divergent, as the information given doesn't allow for a conclusive determination of convergence or the sum.
Geometric series convergence relies on the absolute value of the common ratio (|r|). If |r| is less than 1, the series converges. However, the provided options lack the necessary information about the common ratio to assess convergence accurately.
For instance, if the common ratio were less than 1, options 1), 2), and 3) could be potentially convergent series. Without explicit values for 'r' or 'a', definitively concluding their convergence or sum is impossible. Therefore, the lack of specific terms or common ratios in the options leads to the conclusion that without this information, we cannot determine the convergence or the sum of the series, resulting in the final answer being 4) Divergent. Thus the correct option is 4) Divergent.