Let {a_{0}, a_{1}, a_{2}, ... } be an infinite sequence of real or complex numbers. For n ≥ 0, we define
where it follows that a_{n} = S_{n} − S_{n−1} with a_{0} = S_{0}. Then the infinite series whose nth term is a_{n} is defined to be
and S_{n} is called the nth partial sum of this series. If there exists a finite number S such that
we say that series (1) is convergent and its sum is S, in which case we write
If lim S_{n} does not exist or is infinite, then we call (1) a divergent series. If, on the other hand, there exists a finite number A such that lim A_{n} = A, we say that series (1) is absolutely convergent. When (2) is satisfied, we define the remainder R_{n} to be S − S_{n}, n ≥ 0. Cauchy's convergence principle gives the necessary and sufficient conditions for (1) to be convergent: for any ε > 0 there is an N such that |S_{m} − S_{n} | < ε for all m > n > N. The same holds for absolute convergence by taking A_{m} − A_{n} < ε instead. Absolute convergence implies convergence but not vice versa; the series
is convergent but not absolutely convergent; such series are said to be conditionally convergent. If series (1) converges, then a_{n} → 0 as n goes to infinity; thus, if lim a_{n} ≠ 0 then the series is divergent. However, the converse is not true as indicated by the harmonic series whose nth term is 1/n. There are several convergence tests for an infinite series of form (1):
Comparison test: the series converges absolutely if we can find a convergent series ∑ b_{n} of |a_{n}| ≤ b_{n} nonnegative real terms such that |a_{n}| ≤ b_{n} for all values of n.
Ratio test: if a_{n} ≠ 0 for all n and the ratios |a_{n+1}/a_{n}| ≤ 1 for all n > N, where L < 1 and N are fixed, then the series is absolutely convergent. If |a_{n+1}/a_{n}| > for all n > N, then the series diverges.
Root test: the series converges absolutely if, for fixed L < 1 and N, the roots for all n > N. The series diverges if for all n > N.
The geometric series converges to the sum 1/(1−r) if |r| < 1 and diverges if |r| ≥ 1.
REFERENCES
Hille, E. (1973) Analytic Function Theory, Vol. I, Chelsea Publishing Co., New York.
Kreyszig, E. (1983) Advanced Engineering Mathematics, John Wiley, New York.
References
- Hille, E. (1973) Analytic Function Theory, Vol. I, Chelsea Publishing Co., New York.
- Kreyszig, E. (1983) Advanced Engineering Mathematics, John Wiley, New York.