2. Let xn =n+1n2+1 .(a) Prove that (xn) is a decreasing sequence.(b) Prove that P?n=1(?1)n+1xn is a convergent series.(c) Find a constant c > 0 such that xn ?cnfor all n ? N.(d) Determine whether the series P?n=1(?1)n+1xn is absolutely or conditionally convergent.

3. Let P?n=1 xn and P?n=1 yn be convergent series. Show that:(a) P?n=1(axn) = aP?n=1 xn for any a ? R.(b) Show that P?n=1(xn + yn) = P?n=1 xn +P?n=1 yn.These results are series analogues of the Algebraic limit theorem.(c) Show that the assumption that both series converge is necessary for part (b).(d) Is it true that P?n=1 xnyn =P?n=1 xn P?n=1 yn

6. Study the convergence of the following series:(a) X?n=12nn2(b) X?n=1n22n(c) X?n=1(?1)n+1n2 + 2n2 + 1(d) X?n=2nlog n(log n)n(e) X?n=1?n + 1 ??nn(f) X?n=1(xn+1 ? xn) for any sequence (xn)