Alternating Series Error Bound · 交错级数误差界
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| alternating series error bound/ˈɔːltəneɪtɪŋ ˈsɪəriːz ˈerə baʊnd/ | 交错级数误差界 | jiāo cuò jí shù wù chā jiè |
How close is a partial sum?
- If you stop an alternating series after a few terms, how far off is your estimate?
- For a convergent alternating series, there's a beautifully simple bound.
- The alternating series error bound 交错级数误差界: the error is no bigger than the first omitted term.
- One term tells you the accuracy — no messy computation.
部分和有多接近?
- 若你在几项之后停止一个交错级数,你的估计偏差多少?
- 对收敛的交错级数,有一个美妙而简单的界。
- 交错级数误差界:误差不大于第一个略去的项。
- 一项就告诉你精度——无需繁杂计算。
The bound
- Approximate the sum $S$ by the $n$th partial sum $S_n$. The remainder (error) is $|S-S_n|$.
- For a convergent alternating series (decreasing $b_n\to0$):
-
$$|S-S_n|\le b_{n+1}$$
- The error is at most the absolute value of the next term — the first one you left out.
那个界
- 用第 $n$ 个部分和 $S_n$ 近似和 $S$。余项(误差)是 $|S-S_n|$。
- 对收敛的交错级数(递减 $b_n\to0$):
-
$$|S-S_n|\le b_{n+1}$$
- 误差至多是下一项的绝对值——你略去的第一项。
The next term bounds the error · 下一项界定误差
Partial sums of an alternating series trap the true sum, so the error is at most the size of the next (omitted) term. · 交错级数的部分和将真实和夹在其中，因此误差不超过下一项（被省略项）的大小。
For a convergent alternating series, the error $|S-S_n|$ is at most... · 对于收敛的交错级数，误差 $|S-S_n|$ 至多为...
$|S-S_n|\le b_{n+1}$.
The error is bounded by the first ____ term, not the last included one. · 误差不超过第一个 ____ 项，而非最后一个包含的项。
Use $b_{n+1}$, the term you did not add. · 使用 $b_{n+1}$，即你未加上的那一项。
Why the next term bounds it
- Because terms alternate and shrink, the true sum is always trapped between consecutive partial sums.
- So stopping at $S_n$ overshoots or undershoots by less than the size of the next term.
- The partial sums close in like a shrinking accordion around $S$.
- Hence the very next term is a guaranteed error ceiling.
为何下一项界住它
- 因为各项交替且缩小,真实的和总被夹在相邻部分和之间。
- 所以停在 $S_n$ 的超出或不足小于下一项的大小。
- 部分和像收缩的手风琴一样围着 $S$ 靠拢。
- 因此紧接着的下一项是有保证的误差上限。
Approximating $1-\tfrac12+\tfrac13-\cdots$ with $3$ terms, the error bound is the next term $b_4$. What is it? · 使用 $1-\tfrac12+\tfrac13-\cdots$ 项近似 $3$，误差界为下一项 $b_4$。它是什么？
$b_4=\tfrac14=0.25$.
The true sum is always trapped between consecutive partial sums of an alternating series. · 真实和始终被夹在交错级数的相邻部分和之间。
That is why the next term bounds the error. · 这就是为什么下一项界定误差的原因。
Using it to guarantee accuracy
- Want the error under $0.01$? Find the first term with $b_{n+1}\le 0.01$ and stop there.
- The bound lets you decide how many terms you need for a target accuracy.
- It only applies to alternating series meeting the Alternating Series Test conditions.
- Simple, tight, and exam-friendly.
用它保证精度
- 想要误差小于 $0.01$?找第一个满足 $b_{n+1}\le 0.01$ 的项,停在那里。
- 这个界让你决定为目标精度需要多少项。
- 它只适用于满足交错级数判别法条件的交错级数。
- 简单、紧、考试友好。
This error bound applies to any convergent series, alternating or not. · 此误差界适用于任意收敛级数，无论是否交错。
Only alternating series meeting the test conditions. · 仅适用于满足判别法条件的交错级数。
To guarantee an error under $0.01$, you keep terms until the next term $b_{n+1}$ is... · 为保证误差不超过 $0.01$，需保留项直到下一项 $b_{n+1}$ 是...
Stop when $b_{n+1}\le0.01$. · 当 $b_{n+1}\le0.01$ 时停止。
This bound works only for alternating series that satisfy the Alternating Series Test (decreasing $b_n\to0$) — it does not apply to positive-term series. The error is bounded by the first omitted term $b_{n+1}$, not the last included one. Use $b_{n+1}$, the size of the term you didn't add.
这个界仅对满足交错级数判别法(递减 $b_n\to0$)的交错级数有效——它不适用于正项级数。误差被第一个略去的项 $b_{n+1}$ 界住,而非最后包含的那个。用 $b_{n+1}$,你没有加上的那项的大小。
Estimate $\sum_{n=1}^\infty\tfrac{(-1)^{n+1}}{n}=1-\tfrac12+\tfrac13-\cdots$ with the first $3$ terms; bound the error.
- $S_3=1-\tfrac12+\tfrac13=\tfrac{5}{6}\approx0.833$.
- First omitted term: $b_4=\tfrac14$. So $|S-S_3|\le\tfrac14=0.25$.
- The true sum $\ln 2\approx0.693$ is indeed within $0.25$ of $0.833$. ✓
用前 $3$ 项估计 $\sum_{n=1}^\infty\tfrac{(-1)^{n+1}}{n}=1-\tfrac12+\tfrac13-\cdots$;界定误差。
- $S_3=1-\tfrac12+\tfrac13=\tfrac{5}{6}\approx0.833$。
- 第一个略去的项:$b_4=\tfrac14$。所以 $|S-S_3|\le\tfrac14=0.25$。
- 真实的和 $\ln 2\approx0.693$ 确实在 $0.833$ 的 $0.25$ 以内。✓
The alternating series error bound: for a convergent alternating series, the error of the $n$th partial sum is at most the first omitted term, $|S-S_n|\le b_{n+1}$. It applies only to alternating series meeting the test's conditions, and lets you pick how many terms guarantee a target accuracy.
交错级数误差界:对收敛的交错级数,第 $n$ 个部分和的误差至多是第一个略去的项,$|S-S_n|\le b_{n+1}$。它只适用于满足判别法条件的交错级数,并让你选多少项能保证目标精度。