However, a series converges absolutely if the series continue to converge when the absolute values of all its terms have been taken, that is, all values have become non-negative. When the sequence of partial sums of a series converges, the series converges. The test of convergence of a series follows a series of measures. The radius of convergence is the interval with the values (-R, R). Such series would either converge when the value of x equals zero or for all real values of x, or for all real values of x given that –R Alternating Series: An alternating series is a series in which the terms alternate in sign. For p=2 which is the sum of the inverses of the squares, the sum is (p^2)/6.Ĥ. The series converges if the common ratio is clearly greater than 1 though the value of the sum at this point is only known in a few instances. The P-series diverges if the common exponent is less than or equal to 1 which is in sharp comparison with the harmonic series. P-Series: P-series is a series where the common exponent p is a positive real constant number. If the common ratio is greater than or equal to 1, the series diverges while if the value of the common ratio is between 0 and 1, it converges and the sum is given by:ģ. Geometric Series: Geometric Series is a series where the ratio of each two consecutive terms is a constant function of the summation index.Ī geometric series may converge or diverge depending on the value of the common ratio. However, the alternative harmonic series converges to the natural logarithm of 2.Ģ. Harmonic series is divergent because its sequence of partial sums is rather unbounded. Harmonic Series: This is an example of divergent series. Let the terms in a series be denoted by the symbol, a n, and the nth partial summation be denoted using the following sigma notation for any natural number n:ġ. If the terms of a rather conditionally convergent series are suitably arranged, the series may be made to converge to any desirable value or even to diverge according to the Riemann series theorem. A series contain terms whose order matters a lot. The infinite series often contain an infinite number of terms and its nth term represents the nth term of a sequence. A series is an infinite addition of an ordered set of terms.