Editing Series (mathematics) (section)

===Evaluation of truncation errors===
The evaluation of truncation errors of series is important in [[numerical analysis]] (especially [[validated numerics]] and [[computer-assisted proof]]). It can be used to prove convergence and to analyze [[Rate of convergence|rates of convergence]].

====Alternating series====
{{Main|Alternating series}}
When conditions of the [[alternating series test]] are satisfied by <math display=inline>S:=\sum_{m=0}^\infty(-1)^m u_m</math>, there is an exact error evaluation.<ref>[https://www.ck12.org/book/CK-12-Calculus-Concepts/section/9.9/ Positive and Negative Terms: Alternating Series]</ref> Set <math>s_n</math> to be the partial sum <math display=inline>s_n:=\sum_{m=0}^n(-1)^m u_m</math> of the given alternating series <math>S</math>. Then the next inequality holds:
<math display=block>|S-s_n|\leq u_{n+1}.</math>

====Hypergeometric series====
{{Main|Hypergeometric series}}
By using the [[ratio]], we can obtain the evaluation of the error term when the [[hypergeometric series]] is truncated.<ref>Johansson, F. (2016). Computing hypergeometric functions rigorously. arXiv preprint arXiv:1606.06977.</ref>

====Matrix exponential====
{{Main|Matrix exponential}}
For the [[matrix exponential]]:

<math display=block>\exp(X) := \sum_{k=0}^\infty\frac{1}{k!}X^k,\quad X\in\mathbb{C}^{n\times n},</math>

the following error evaluation holds (scaling and squaring method):<ref>Higham, N. J. (2008). Functions of matrices: theory and computation. [[Society for Industrial and Applied Mathematics]].</ref><ref>Higham, N. J. (2009). The scaling and squaring method for the matrix exponential revisited. SIAM review, 51(4), 747-764.</ref><ref>[http://www.maths.manchester.ac.uk/~higham/talks/exp10.pdf How and How Not to Compute the Exponential of a Matrix]</ref>

<math display=block>T_{r,s}(X) := \biggl(\sum_{j=0}^r\frac{1}{j!}(X/s)^j\biggr)^s,\quad \bigl\|\exp(X)-T_{r,s}(X)\bigr\|\leq\frac{\|X\|^{r+1}}{s^r(r+1)!}\exp(\|X\|).</math>