Series
         Absolute convergence
         Conditional convergence
      Series of positive terms
Absolute convergence of a series
An infinite series is called absolutely convergent if the infinite series is convergent, i.e.,
the series of absolute values of its terms converge.
Thus, if a series is absolutely convergent, then both series,
Example:  Let show that the series   is absolutely convergent.
Solution:  Given series is absolutely convergent since the series of absolute values of its terms
  converges to 2,  
Therefore, the series is convergent and its sum is 2/3.
Example:  Let show that the series   is absolutely convergent.
Solution:  Given series is absolutely convergent since the series of absolute values of its terms
Therefore, the series converges and its sum
Conditional convergence
If an infinite series is not convergent, it is called divergent. However, if it is convergent but not
absolutely convergent, it is called conditionally convergent.
Series of positive terms
Suppose the series consists of positive terms only. Then its partial sums form an
increasing sequence, therefore given series converges if and only if the sequence of partial sums {sn} is
bounded.
We use this property for convergence tests of series with positive terms.
Example:  The series of positive terms
converges since
that is, the sequence of partial sums is bounded above, and the series converges to the number e.
If we calculate the partial sum of first 10 terms of the series, s10 = 2.718281 obtained is the number e to six correct decimal digits what shows that the series increases relatively fast.
Functions contents E
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.