remainder term series
For any series
${a}_{1}+{a}_{2}+{a}_{3}+\mathrm{\dots}$ | (1) |
of real or complex terms ${a}_{j}$ one may interpret its $m$’th remainder term
${R}_{m}:={a}_{m+1}+{a}_{m+2}+\mathrm{\dots}$ | (2) |
as a series. This remainder term series has its own partial sums
${S}_{m}^{(n)}:={a}_{m+1}+{a}_{m+2}+\mathrm{\dots}+{a}_{m+n}\mathit{\hspace{1em}\hspace{1em}}(n=\mathrm{\hspace{0.33em}1},2,\mathrm{\dots}).$ | (3) |
If $m+n=k$, then the ${k}^{\mathrm{th}}$ partial sum of the original series (1) is
${S}_{k}={S}_{m}+{S}_{m}^{(n)}.$ | (4) |
For a fixed $m$, the limit ${lim}_{k\to \mathrm{\infty}}{S}_{k}$ apparently
exists iff the limit
${lim}_{n\to \mathrm{\infty}}{S}_{m}^{(n)}$ exists. Thus we can write the
Theorem. The series (1) is convergent^{} if and only if
each remainder term series (2) is convergent.
Cf. the entry ‘‘finite changes in convergent series’’.
References
- 1 Л. Д. Кудрявцев: Математический анализ. Издательство ‘‘Высшая школа’’. Москва (1970).
Title | remainder term series |
---|---|
Canonical name | RemainderTermSeries |
Date of creation | 2014-05-16 21:09:46 |
Last modified on | 2014-05-16 21:09:46 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 13 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 40-00 |