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Series with Interval of Convergence Dependent on Structure

​
x
0.280776
number of terms
100
series
3x-
5
2
x
2
+144+
70252111
11
-
1267650600228514967032053771
100
difference in sum
{0.158,-0.145,0.122,-0.102,92,0.005,-0.005,0.005,-0.005}
plot
This Demonstration shows an infinite series that, when written in two different ways, has a different interval of convergence.
The power series expansion of
log(1+3x+2
2
x
)
is
3x-
5
2
2
x
+3
3
x
-
17
4
4
x
+…
. The range of validity of the expansion, for which the series converges, is
-0.5<x≤0.5
. However, if the series is written in the form
log(1+u)=u-
2
u
2
+
3
u
3
-
4
u
4
+…
, where
u=3x+2
2
x
, that is,
(3x+2
2
x
)-
2
(3x+2
2
x
)
2+
3
(3x+2
2
x
)
3-
4
(3x+2
2
x
)
4+…
, the interval of convergence is reduced to
-0.5<x≤0.280776
.
This Demonstration shows the difference between these two forms (the blue and purple graphs, respectively). It shows that beyond the value of
x=0.280776
, the second series diverges (the corresponding graph now changes color to red). The aim is to emphasize the fact that even though the first few terms and the sum to infinity (shown in green) of two infinite series may coincide, the actual structure of the series makes a difference with regard to the radius/interval of convergence.
(The notation <<
m
>> indicates
m
omitted terms.)
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