Homework 4: Do all of the following problems. Make sure
not to skip steps.
1. Recall that for an infinite series
โ
X
k=1
ak
we say that the sum converges if the sequence of partial sums
Sn=
n
X
k=1
ak=a1+a2+ยท ยท ยท +an
converges. In this case we say that
โ
X
k=1
ak= lim
nโโ
Sn.
Also recall the following property of limits. If rnand tnare two se-
quences then if they both converge we get that the sequence vn=rn+tn
also converges and
lim
nโโ
vn= lim
nโโ
rn+ lim
nโโ
tn.
Use this to show that if
โ
X
k=1
ak,and
โ
X
k=1
bk
converge then
โ
X
k=1
(ak+bk)
converges and
โ
X
k=1
(ak+bk) =
โ
X
k=1
ak+
โ
X
k=1
bk.
2. Determine if the following series converge or diverge. If they
converge give a bound for the sum. The integral test and the geometric
series formula will be useful.
(a)
โ
X
k=1
1
kln(k)2;
1
