# MATH 1572H Spring 2013 Worksheet 4 Topic: limit of a sequence

```MATH 1572H
Spring 2013
Worksheet 4
Topic: limit of a sequence; basic properties of series.
Two useful facts
• For any sequence {an }, which goes to infinity, lim 1 +
• For any sequence {bn }, which converges to zero, lim
1
an
sin bn
bn
an
= e.
= 1.
1. For each of the given sequences, find its limit.
1 3n
a) lim 1 + 2n
b) lim
n+1 n+3
n+2
sin
3
c) lim sin n2
n
sin
d) lim √ 1
n
1
n
+1−1
Series
Definition. Let {an } be a sequence. Then the expression of the form
a1 + a2 + · · · + an + . . . is called a series.
We attach a numerical value (”the sum“) to a series
∞
P
an or, simply,
n=1
∞
P
an by forming a the sequence of
n=1
partial sums
s 1 = a1 ,
s2 = a1 + a2 ,
s3 = a1 + a2 + a3 ,
sn = a1 + a2 + · · · + an , . . .
and then finding its limit s = lim sn . It may happen that the limit does not exist. In such a
case, we just say that the given series diverges.
1. For each of the given series, find a formula for its n-th partial sum sn and then find the
sum of the series (if it exists).
a) 1 + 0.1 + 0.01 + 0.001 + . . .
b) 4 − 2 + 1 − 21 + 14 − 18 + . . .
c)
∞
P
n=1
3n +2n
5n
d) x + 2x2 + 3x3 + 4x4 + . . .
e)
1
2·4
+
1
3·5
+
1
4·6
+ ··· +
1
n(n+2)
+ ...
2
```