Graphics Reference
In-Depth Information
s
, and
s
17 If
a
is the
n
th partial sum of this series, find a
valuefor
a
and provethat this tends to 0 as
n
.
18 Write down the claim that a sequence of partial sums (
s
)isa
Cauchy sequence. From the General Principle of Convergence (qn
4.57) this is equivalent to its convergence. Interpret this claim for
the infinite series from which the partial sums were formed.
19
The scalar rule
For a given real number
c
0, provethat
c
·
a
is convergent if
and only if
a
is convergent.
20 Use questions 9 and 19 to determine exactly which geometric series
are convergent and which are divergent.
21
The sum rule
If
a
and
b
are convergent, prove that
(
a
b
) is convergent.
22
(a) If
a
is convergent, is
a
convergent?
(b) If
s
is the
n
th partial sum of the series
a
a
a
a
...
which of thefollowing is equal to
s
...
?
(i)
, (ii)
, (iv)
a
a
, (iii)
a
a
.
Summary
-
convergence of series
Definition of convergence of series
For any sequence (
a
), the
n
th partial sum
s
a
may be constructed. When (
s
)
s
,
the series
) is said
to be
convergent
and to converge to the sum
s
.A
series which is not
convergent
is said to be
divergent
.
a
(or briefly
a
The start rule
qn 16
a
is convergent if and only if
a
is convergent.
T
he scalar rule
qn 19
a
is
convergent, for some non-zero constant
c
.
is convergent if and only if
c
·
a
The sum rule
qn 21
If
a
and
b
are convergent, then
(
a
b
)
is convergent.
