结果1 题目 Show that the Maclaurin series for ln (1+x) is x- (x^2)2+ (x^3)3-⋯ + ((-1)^(n+1)x^n)n+⋯ . This series is valid for -1 x≤ 1 only; by drawing graphs of y=ln (1+x) and several successive approximations show that this is plausible. (This ...
【题目】Show that th e Maclaurin series for $$ ( 1 + x ) ^ { n } $$ e is$$ 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \cdots $$$ + \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { r ! } x ^ { r } + \cdots $$ ...
+ ... where f^(n)(0) is the n-th derivative of f(x) evaluated at 0, and 'n!' is the factorial of n. What is the Maclaurin series used for? The Maclaurin series is used for approximating functions near the origin, analyzing the behavior of a function, solving problems in various...
xnf(x)=f(0)+f′(0)1!x+f″(0)2!x2+⋯+f[n](0)n!xn subject to the conditions holding for a Taylor series called also Maclaurin's series Word History Etymology Colin Maclaurin †1746 Scottish mathematician First Known Use 1881, in the meaning defined above Time Traveler The ...
(1) Maclaurin series are named after the Scottish mathematician Colin Maclaurin. The Maclaurin series of a function up to order may be found using Series[f, x, 0, n]. The th term of a Maclaurin series of a function can be computed in the Wolfram Language using SeriesCoefficient[f, ...
So we need the first five (nonzero) terms of the Maclaurin series for the desired accuracy. (In fact, this sum is approximately 0.33698 and ln1.4≈ 0.33647.)结果一 题目 【题目】How many terms of the Maclaurin series forn(1+z) do you need to use to estimate n1.4 towithin 0.001?
Your plan is to use the well- known Maclaurin series for cos x :cos x=∑limits ^(∞ )_(n=0) ((-1)^nx^(2n))((2n)!), (3)First, note that cos ^2(x)= (1+cos (2x))2. Therefore, replacing x by 2x in Eq. (3), you obtain (1+cos (2x))2= 12+ 12∑limits _(n...
Use the Maclaurin series {eq}\ln(1 + x) = \sum\limits_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} {x^n} {/eq} to find the Maclaurin series of {eq}\ln(1 - x). {/eq} Maclaurin Series of a Function The Maclaurin ...
Find the Maclaurin series up to the term in x3 for the function f(x)=(1+x)1/21−x. Solution As this function would be difficult to differentiate three times (to use the Maclaurin series directly), we use f(x)=(1+x)1/2(1−x)−1 and find series for the two terms in the...
Show how to find the first few terms of the Maclaurin series for x1+x?Question:Show how to find the first few terms of the Maclaurin series for x1+x? Maclaurin:Finding the Maclaurin series of a one-variable infinitely differentiable function is equivalent to finding t...