Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
STEP_BY_STEP SOLUTION Step-1 Definition ; A series is said to be convergent if it approaches some limit .Formally , the infinite series a n is convergent if the sequence of partial sums n=1 n S n a k is convergent . Conversely , a series is divergent if the k=1sequence ofpartial sums is divergent.NOTE : The terms grow without bound , so the sequence does not converge. If U k and V k are convergent series , then ( U + Vk k ) and ( U - V k k ) areconvergent . If C = / 0 , then U k and V k both converge or both diverge . Step-2 (1) 1 nThe given sequence is a = n n = ( 0.9 (0.9) n = ( 10) , since 0.9 = 9 9 10 = ( 1) ( ) 10 n …………….(1) 9 10 10 n Here , 9 >1 . So , as n , then the value of ( ) 9 also tending to infinity. From the step (1) , we know that , The terms grow without bound , so the sequence does notconverge. n (1) n 10 n Therefore , lna =nlim n (0.9) = lin 1) ( ) 9 , since from (1).\n Therefore , the given sequence does not converge , since as n ,the termsgrow without bound . (1)n Therefore , lim a = lnm n does not converge. n n (0.9)