2.4. Basic theorems
Theorem 1. The sum of a uniformly convergent series whose terms are continues functions cannot have discontinues.
The
proof. Indeed,
is continuos as a sum of finite number of continues functions. If the
series is uniformly converging its remainder
will be arbitrarily small over the whole interval
for sufficiently large values of
.
Therefore a small variation of
yields small variations both of
and
,
and thus the whole sum
gains a small increment as well which means that the sum cannot have
discontinuities.
Remark. The functional series whose terms are continuous on but which does not uniformly converge on may have as its sum a discontinuous function.
Example 2.7. Consider the functional series
Solution
We take its th partial sum:
Therefore
i.e.
the sum
.
It is discontinuous on the interval
,
although the terms are continuous. The original series does not
converge uniformly on the uniformly on the interval
Theorem
2. A
uniformly convergent series can be integrated term-wise i.e. for any
and x from the interval
we have
Where
the series thus obtained is uniformly converging on
The proof. In fact,
Theorem 3. A uniformly convergent series whose terms are continuous differentiable functions can be differentiated, if after that we obtain uniformly convergent series and
if
To prove the statement we denote the sum of the series
.
Then integrating term-by-term we arrive to the equality:
Finally,
differentiating the last relation, we obtain:
2.5. Power series
Functional
series of the form
,
where
are given real numbers, and
is a real variable, are called power
series.
We can easily find out for what numerical values of the power series converges. Since
Where
(of course we suppose that the limit exists), we conclude, on the basic of ratio test, that the series is absolutely convergent for
The
interval
is the
interval of convergence of
power series, and
is called the radius
of convergence.
Abel’s
theorem. If
a power series
is
convergent at a point
then it converges absolutely at any point
such that
.
But if the series diverges at any point
,
then it diverges at any point
for which
.
From Abel’s theorem it follows that for a power series three cases are possible:
Series converges only at the point ;
Series converges at all
;There exists a positive number , such that for all from interval
series converges, and for all
the
series diverges.
According
to ratio test, if the limit
exists, we have
.
The limit does not exist in some cases, but then the radius of convergence can sometimes be found by means of root test:
Example 2.8. Find the domain of convergence of the series
Solution
Since
,
we get
Thus,
the series converges on the interval
.
The number series
converges,
the given series converges absolutely for
.
Finally, the given series converges on the interval
and diverges for all
.
Example
2.9. Find
the domain of convergence of the series
Solution
At
first we find the radius of convergence:
so that we have the interval
.
Since at the points
the given series turns into divergent series of the form
Finally , the given series converges absolutely on the interval
and diverges for all
