Convergent Series

The infinite Series $\sum_{n=1}^\infty a_n$ is convergent if the Sequence of partial sums

$$S_n = \sum_{k=1}^n a_k$$

is convergent. Conversely, a Series is divergent if the Sequence of partial sums is divergent. If $\sum u_k$ and $\sum v_k$ are convergent Series, then $\sum (u_k+v_k)$ and $\sum (u_k-v_k)$ are convergent. If $c\not=0$, then $\sum u_k$ and $c\sum u_k$ both converge or both diverge. Convergence and divergence are unaffected by deleting a finite number of terms from the beginning of a series. Constant terms in the denominator of a sequence can usually be deleted without affecting convergence. All but the highest Power terms in Polynomials can usually be deleted in both Numerator and Denominator of a Series without affecting convergence. If a Series converges absolutely, then it converges.

See also Convergence Tests, Radius of Convergence

References

Bromwich, T. J. I'a. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991.