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SERIES EXPANSIONS

DOI: 10.1615/AtoZ.s.series_expansions

Let {a0, a1, a2, ... } be an infinite sequence of real or complex numbers. For n ≥ 0, we define

where it follows that an = Sn − Sn−1 with a0 = S0. Then the infinite series whose nth term is an is defined to be

(1)

and Sn 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

(2)

If lim Sn 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 An = A, we say that series (1) is absolutely convergent. When (2) is satisfied, we define the remainder Rn to be S − Sn, 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 |Sm − Sn | < ε for all m > n > N. The same holds for absolute convergence by taking Am − An < 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 an → 0 as n goes to infinity; thus, if lim an ≠ 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):

  1. Comparison test: the series converges absolutely if we can find a convergent series ∑ bn of |an| ≤ bn nonnegative real terms such that |an| ≤ bn for all values of n.

  2. Ratio test: if an ≠ 0 for all n and the ratios |an+1/an| ≤ 1 for all n > N, where L < 1 and N are fixed, then the series is absolutely convergent. If |an+1/an| > for all n > N, then the series diverges.

  3. 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.

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