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
Properties of series
The product of two series or the Cauchy product
The remainder or tail of the series
 As with sequences, the convergence of an infinite series only depends on the behavior of the
general term of the series an as n increases to infinity, and not on any finite number of its initial terms.
 Note that, since
 the series converges if and only if converges.
 Therefore, to show that a series converges we can ignore any finite number of terms at the
 beginning, and just need to prove the convergence of the tail or remainder of the series.
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 converges then the remainder rn  = an + 1 + an + 2 + an + 3 + . . .  converges too,
 that is, since and  s  = sn + rn  then,
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 converges is, that for given however small positive number e it is possible find
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
| an + 1 + an + 2 + an + 3 + . . . + an + r  | < 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.
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
 increases to infinity, that is So, if this condition is not satisfied the series diverges.
That this condition is only necessary but not sufficient condition for the convergence shows the harmonic series for which
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.
The nth term test for divergence
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.
Properties of series
If given are two convergent series,
then the convergent series is obtained by adding or subtracting their same index terms, and its sum equals the sum or the difference of their individual sums, i.e.,
The product of two series or the Cauchy product
 If given are two convergent series of positive terms, then the product
denotes the convergent series sum of which is equal to the product of the sums of the given series.
Calculus contents B