Examples of using 级数 in Chinese and their translations into English
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艾萨克·牛顿利用无穷级数计算π到第15位，后来写道：“我很羞愧的告诉你我为了这个计算用了多少个数字。
Isaac Newton used infinite series to compute to 15 digits, later writing"I am ashamed to tell you to how many figures I carried these computations".以现代的觀點看，级数1-2x+3x2-4x3+…并没有定义一个在x=1时的函数，因此不能简单地把值代入到其相應的表达式。
In the modern view, the series 1- 2x+ 3x2- 4x3+… does not define a function at x= 1, so that value cannot simply be substituted into the resulting expression.虽然在标准的表示法中，交错调和级数收敛于ln(2)，我们可以把它的项重新排列，使它收敛于任何一个数，甚至发散。
Although in standard presentation the alternating harmonic series converges to ln(2), its terms can be arranged to converge to any number, or even to diverge.X的指数，用eX或exp(X)来表示，是由以下幂级数所给出的n×n矩阵：eX=∑k=0∞1k!
The exponential of X, denoted by eX or exp(X), is the n×n matrix given by the power series e X=∑ k= 0∞ 1 k!为了检查级数是否收敛于f(x){\displaystylef(x)}，通常采用泰勒定理估计级数的餘项。
To check whether the series converges towards f( x){\displaystyle f(x)}, one normally uses estimates for the remainder term of Taylor's theorem.艾萨克·牛顿用无穷级数计算π到15位数字，后来写作“我很惭愧地告诉你多少数字我进行这些计算”。
Isaac Newton used infinite series to compute to 15 digits, later writing"I am ashamed to tell you to how many figures I carried these computations".柯西最终给出了（收敛）级数的和的严格定义，从这过后的一段时间，发散级数基本被排除在数学之外了。
Augustin-Louis Cauchy eventually gave a rigorous definition of the sum of a(convergent) series, and for some time after this, divergent series were mostly excluded from mathematics.合国环境规划署（环境署）工作人员要求审核其晋升后的级数，及要求因福利延迟支付造成财政损失获得赔偿）.
(Claims by United Nations Environment Programme(UNEP) staff member for review of step level upon promotion, and for compensation for financial losses resulting from late payment of entitlements).例如，要得到ƒ(0.1)=sin(0.1)的前5位有效数字，只需要计算级数的前两项。
For example, if we want to calculate f(0.1)= sin(0.1) accurate up to five decimal places, we only need 泰勒级数.
Taylor Series.幂级数表示.
Power series expression.这些避免依赖无穷级数。
These avoid reliance on infinite series.泰勒级数的扩展是：.
The Taylor series expansions are.泰勒级数的扩展是：.
Taylor series expansions around are.泰勒级数的扩展是：.
The full Taylor series expansion is.级数的部分和partialsum.
Series and Partial Sums.级数：行星齿轮组的数量。
Series: the number of sets of planetary gears.Video:什么是几何级数？?
这个被修改后的级数被称为Kempner级数。
This modified harmonic 那么函数的Fourier级数就定义为.
Where the coefficients of Fourier series are defined as.
