Answer and Step-by-step explanation:
It seems like you’re asking about the convergence of several sequences. Here are my answers:
(a) The sequence a_n = (n+1)(-1)^n/n is an alternating sequence. The limit of the absolute value of the sequence |a_n| = (n+1)/n is 1, which is not equal to 0. Therefore, by the Alternating Series Test, the sequence a_n does not converge.
(b) The sequence b_n = ((n+1)/n - n/n)^n can be simplified to b_n = (1 + 1/n)^n. This sequence converges to Euler’s number e ≈ 2.71828.
© The sequence c_n = ((n+1)/n - n/n)^n can be simplified to c_n = (1 + 1/n)^n. This sequence converges to Euler’s number e ≈ 2.71828.
(d) The sequence d_n = (3^n)^(1/(2n)) can be simplified to d_n = 3^(1/2). Since this expression does not depend on n, the sequence converges to 3^(1/2).
(e) For the sequence e_n = (a^n + b^n)/(a^(n+1) + b^(n+1)), where 0 < a < b, we can use L’Hopital’s Rule to find that the limit of the sequence is equal to the limit of (na^(n-1)b^n)/(a^nb^(n-1)(n+1)), which simplifies to (nb)/(a(n+1)). As n approaches infinity, this expression approaches 0. Therefore, the sequence converges to 0.