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To examine the convergence of the given series, let's analyze them numerically and apply standard convergence tests. --- ### (i) Given series: \[ S = \frac{3\times4}{5^2}+\frac{5\times6}{7^2}+\frac{7\times8}{9^2}+\cdots \] #### Step 1: General Term Identification Observing the pattern in the numerators and denominators, the general term can be written as: \[ a_n = \frac{(2n+1)(2n+2)}{(2n+3)^2} \] where \( n \) starts from 1. #### Step 2: Limit Test for Convergence To check whether the series ___ _________ ____ __________ ___ ________ ______ _______ ___ ________.
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