Word Count : 289

---

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 ____ _______ _________ ______ ____ _______ __________.
___ _______ ______ _____ _______.
_____ _______ ______ ______ ____ _______ _____.
____ ________ __________ _____ _________ _____ __________ __________ __________ _________ _______ ____.
___ __________ ________ ____ _______ _______ _______ __________ _________.
________ _______ __________ _______ ____ ___ _________ ____ _____.
______ ______ _____ _____ _______ _______ _____ ____ ______ ___.
____ ___ __________ ______ ____ ____ _______ _______ _________ ___.
__________ _________ _________ ___ _________.
________ ___ ____ ______ __________ _________.
__________ _______ ______ ___ ________ __________ _______ __________ ________.
_______ ____ ___ ____ ______ ____ _________ __________ _________ __________.
_____ __________ ______ ___ _______ ___ _______.
_______ _______ ________ __________ ___ ________ _____ _____ _______.
_______ _________ _______ ______ _________ ______ ________ _______ __________ ______ _______.
_____ __________ _____ ________ _________ ___ _______ __________ ______ ____ _______ ___.
____ _______ _____ _________ ____ __________.
________ ___ ______ _______ ______ ___ _______ ________ ________ ____.
____ ______ _____ ___ ______ _____ ___ ____ ___ ____ ______.
___ _________ __________ _______ _____ __________ _____.
______ _______ ___ _________ _______.
____ ______ ___ _________ ___ _________ ____ _____.
_______ _____ _________ __________ ______ ___ ______ _____ ______ ________ ______ __________.
_________ _____ ______ _______ ____ ____ _________ ______.
______ ____ ____ ____ ________ __________ _______ ____ ______ _______ ________.
_________.
Get Full Answer on WhatsApp