Mathematics Homework Help

2. Let xn =
n+1
n2+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 ≥
c
n
for 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) = a
P∞
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=1
2
n
n2
(b) X∞
n=1
n
2
2
n
(c) X∞
n=1
(−1)n+1n
2 + 2
n2 + 1
(d) X∞
n=2
n
log n
(log n)
n
(e) X∞
n=1
√
n + 1 −
√
n
n
(f) X∞
n=1
(xn+1 − xn) for any sequence (xn)