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Series |
Infinite series |
The remainder
or tail
of the series |
Necessary
and sufficient
condition for the convergence of a series |
Necessary
condition for the convergence of a series |
The
n-th
term test for divergence of series |
Properties
of series |
The
product of two series or the Cauchy product |
Geometric
series |
P-series |
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The remainder
or tail
of the series |
As
with sequences, the convergence of an infinite series |
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only
depends on the behavior of the |
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general
term of the series an
as n
increases to infinity, and not on any finite number of its initial
terms. |
Note
that, since |
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the series |
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converges
if and only if |
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converges. |
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Therefore,
to show that a series |
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converges we can ignore any finite number of
terms |
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at the |
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beginning,
and just need to prove the convergence of the tail or remainder |
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of
the series. |
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The
difference between the sum s
of a convergent series a1
+
a2
+ a3
+
. . . + an
+
. . .
and the nth
partial sum sn
is called the remainder (tail) rn
of the series, i.e., |
rn
= s -
sn
= an + 1 + an + 2 +
an + 3 +
. . .
or
s = sn
+ rn. |
Thus,
if a series |
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converges
then the remainder rn
= an + 1 + an + 2 +
an + 3 +
. . . converges
too, |
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that
is, since |
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and
s
= sn
+ rn
then, |
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Necessary
and sufficient
condition for the convergence of a series - Cauchy's convergence test |
Necessary
and sufficient
condition that the sequence of partial sums {sn}
of a given series converges, and |
hence
the
series |
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converges, is
that for given however small positive number e
it is possible to find |
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an index n0 such
that |
sn
+ r -
sn
| < e
whenever
n
> n0
(e)
and r =
1, 2, 3,
. . . , |
or
expressed by terms of the series, if |
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an + 1 + an + 2 +
an + 3 +
. . . +
an + r
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< e
whenever
n
> n0
(e)
and r = 1, 2, 3,
. . . |
Therefore,
a series converges if the absolute value of the sum of any finite number
of sequential terms can become
arbitrary small by starting the addition from a term which is far
enough. |
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Necessary
condition for the convergence of a series |
Hence, it is necessary
condition for the convergence of a series that its terms tend to
zero as n
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increases to infinity,
that is |
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So,
if this condition is not satisfied the series diverges. |
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That
this condition is only necessary but not sufficient
condition for the convergence shows the harmonic
series for which |
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as
was shown in previous section. |
Necessary
condition for the convergence of a series is usually used to
show that a series does not converge. |
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The
nth
term test for divergence |
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Note
that this is only a test for divergence. That is, if we can prove that
the sequence {an}
does not |
converge
to 0,
then the infinite series does not converge. |
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Properties
of series |
If
given are two convergent series, |
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then convergent is the series obtained by adding or subtracting their
same index terms, and its sum equals the
sum or the difference of their individual sums, i.e., |
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The
product of two series or the Cauchy product |
If
given are two convergent series of positive terms, |
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then the product |
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denotes the
convergent series sum of which is equal to the product of the sums of
the given series. |
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Geometric
series |
A
series, whose successive terms differ by a constant multiplier, is called a geometric series
and written as |
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If
| x | < 1 ;
the nth
partial sum is |
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Thus, the geometric series is convergent if
| x | < 1
and its sum is |
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If
| x | > 1
then the geometric series
diverges. |
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Example: Show
that the series |
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converges. |
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Solution:
Given
is the geometric series subsequent terms of which are multiplied by the
factor 1/2. |
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Example: Let
prove
that the pure recurring decimal 0.333
... converges to 1/3.
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Solution:
Given
decimal can be written as |
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Example: Let
calculate the square of the convergent
geometric series
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using
the Cauchy product shown above. |
Solution:
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= 1 + 2x + 3x2
+ 4x3 + · · · + (n + 1)xn + · · · |
the
obtained series |
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converges
for 0 < x
<1. |
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P-series |
The
series |
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converges
if
fixed constant
p
> 1 and diverges if p
< 1. |
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By
grouping terms |
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where
the number of terms in parentheses form the sequence 2,
4, 8, ... 2r-1,
... such that |
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therefore,
each value in parentheses is smaller than the corresponding term of the geometric
series |
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Thus,
if p
> 1
then q
< 1,
the geometric series converges so that the given series is also
convergent. |
Euler
discovered and revealed sums of the series for
p
=
2m,
so for example |
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If
p < 1
then np
< n or 1/np
> 1/n, therefore the
terms of the given series are not smaller than the terms |
of
the divergent harmonic series so, given series diverges. |
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