determine the following limits (1) lim n->? (-1)^n / (n^2+1) (2)lim n->? n^1/2((

Question

determine the following limits (1) lim n->? (-1)^n / (n^2+1) (2)lim n->? n^1/2((n+1)^1/2-n^1/2) (3)lim n->? (2n^2+3)/(n^2+2) (4) lim n->? n^(1/n^2) (5) lim n->? (1+2+3+...+n)/n^2 (6) lim n->? (1+3+5+...+(2n-1))/n^2 (7)lim n->? 1+2^m+...+n^m for all m belong to natural number, m?2

Explanation / Answer

lim n->inf (-1)^n / (n^2+1) = lim n->inf [((-1)^n)/(n^2)] / ((1/n^2)+1) = 0 lim n->8 n^1/2((n+1)^1/2-n^1/2) = lim n->8 (n^1/2)((n+1)^1/2)(1 -(n/(n+1))^1/2) = 0 lim n->8 (2n^2+3)/(n^2+2) = lim n->8 (2+3/n^2)/(1+2/n^2) =2 lim n->8 n^(1/n^2) = lim n->8 n^(1/n^2) =1 lim n->8 (1+2+3+...+n)/n^2 = lim n->8 n(n+1)/2n^2 = lim n->8 (1+1/n)/2 = 1/2 lim n->8 (1+3+5+...+(2n-1))/n^2 = lim n->8 [(n/2)(2+2(n-1))]/n^2 = 1 lim n->8 1+2^m+...+n^m = inf. (diverging).

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