Wednesday, February 29
−
Lecture 23 :
Error bounds for
series approximations.
(Refers to Section 8.2 and 8.3 in your text)
After having practiced the problems associated to the concepts of this lecture the student should be
able to
:
Approximate the value of a series converging by the Alternating series test and give a
reasonable upper bound for the approximation.
23.0
Introduction
−
Once we know that a series of numbers converges the next step is to
determine its value. If this cannot be done we can always approximate its value by
truncating the series to one of its partial sums.
In the case of a
geometric series
we have a formula for its limit. So no approximation
is needed. We have a method which allows us to obtain its precise value.
In most other cases our
only
option is to estimate the infinite
sum by computing a
particularly large partial sum
.

When we approximate the value
S
of a series with a partial sum
S
n
, the exact error
is given by the expression
S
–
S
n
. Often the best we can do is to obtain an upper
bound for the value of
S
–
S
n
. This can be sometimes difficult to do depending on
the complexity of the series.

Alternating series have a builtin “error estimation tool”. We see this in the
following theorem.
23.1
Theorem
−
Alternating series error estimation theorem.
If an alternating series
satisfies the two conditions required for convergence then an error bound for the
approximation of its value
S
is given by

R
n
 = 
S
−
S
n

≤
c
n
+ 1
.
where
S
n
represents its
n
th
partial sum. (The symbol
R
n
is referred to as the
remainder
or
the
error
we obtain when we approximate the value of
S
by
S
n
.
Proof:
We have seen in the
Alternating series test
proof that the number
S
is always above
S
2
n
and below
S
2
n +
1
for any value of
n
.
So
S
is always between two successive terms
S
2
n
and
S
2
n +
1
. For example S would between above
S
6
and below
S
7
, below
S
11
but
above
S
10
.
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View Full DocumentSo if we approximate the value of
S
with an “even” partial sum
S
2
n
an upper bound for
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 Winter '08
 ZHOU
 Calculus, Approximation, Mathematical Series, 2m, ∞

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