does the sequence converges or diverges and if it converges what is the limit?

Question

Can someone plz help with this prob. thanksssss
a sub n= (-1)^(n-1)n/ n^(2) +1

Answer 1

Notice that the terms of your sequence are alternately positive and negative. Now consider the function of a real variable defined by f(x) = x/(x^2 + 1). The limit as x -> inf is 0. All the positive terms in your sequence are embedded in f(x), so the positive terms have limit 0.

Next consider the function g(x) = -x/(x^2 + 1). The limit as x -> inf is 0. The negative terms of your sequence are embedded here, so the negative terms have limit 0.

Since both subsequences have the same limit, the entire sequence is convergent, with limit 0.