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Approximatingsquarerootsusingbinomialexpansion.Throughthebinomialexpansionof(1−2x)12,Iamrequiredtofindanapproximationof√2.(1+x)n=1+n1x+n(n−1)1∗2x2+...Thus,theexpansionof(1−2x)12:=1−x−12x2−12x3+...,Hint.Notethatfor|z|<1,(1+z)1/2=∞∑n=0(1/2n)zn=1+z2−z28+o(z2).Nowfora≠0,thesecondorderexpansionat0is ...
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Approximating square roots using binomial expansion. | square root expansion
Approximating square roots using binomial expansion. Through the binomial expansion of (1−2x)12, I am required to find an approximation of √2. (1+x)n=1+n1x+n(n−1)1∗2x2+... Thus, the expansion of (1−2x)12: =1−x−12x2−12x3+... Read More
Binomial expansion of square root | square root expansion
Hint. Note that for |z|<1, (1+z)1/2=∞∑n=0(1/2n)zn=1+z2−z28+o(z2). Now for a≠0, the second order expansion at 0 is ... Read More
Series expansion of square root function | square root expansion
HINT : Rewrite √4−3x2=(4−3x2)12=(4(1−34x2))12=2(1−34x2)12=2(1−(√32x)2)12. Now, let y=√32x and use binomial series for (1−y2)12. Read More
Using the binomial expansion to approximate a square root ... | square root expansion
Here we look at some arguments why taking out a factor 2.5 is a convenient choice. Problem: We want to (manually) approximate √6 by using the first few terms ... Read More
formal power series expansion for square root | square root expansion
The Chu-Vandermonde Identity gives that n∑k=0(1/2k)(1/2n−k)=(1n). Computing the square of the series using (1) and the product formula for ... Read More
Square root | square root expansion
As decimal expansions The square roots of the perfect squares (0, 1, 4, 9, 16, etc.) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore their decimal representations are non-repeating decimals. Read More
Methods of computing square roots | square root expansion
跳到 Continued fraction expansion - Methods of computing square roots are numerical analysis algorithms for finding the principal, or non-negative, ... Read More
Binomial approximation | square root expansion
The binomial approximation is useful for approximately calculating powers of sums of 1 and a ... and recalling that a square root is the same as a power of one half. 1 a + b − 1 a ... For example calculating the multipole expansion. Griffiths, D. Read More
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