1+ x)^- 1/ 2 taylor expansion

22.12.2020 Comments

In mathematicsthe Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.

Taylor's series are named after Brook Taylor who introduced them in If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin seriesafter Colin Maclaurinwho made extensive use of this special case of Taylor series in the 18th century. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations.

If the Taylor series of a function is convergentits sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval or open disk in the complex plane containing x.

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This implies that the function is analytic at every point of the interval or disk. In the more compact sigma notationthis can be written as. The Taylor series for any polynomial is the polynomial itself. The Maclaurin series for the exponential function e x is. The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; [2] the result was Zeno's paradox.

Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Archimedesas it had been prior to Aristotle by the Presocratic Atomist Democritus. It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result.

In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. The Kerala School of Astronomy and Mathematics further expanded his works with various series expansions and rational approximations until the 16th century. In the 17th century, James Gregory also worked in this area and published several Maclaurin series.

It was not until however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor[7] after whom the series are now named. The Maclaurin series was named after Colin Maclaurina professor in Edinburgh, who published the special case of the Taylor result in the 18th century. Thus for x in this disk, f is given by a convergent power series.

Thus a function is analytic in an open disk centred at b if and only if its Taylor series converges to the value of the function at each point of the disk. The polynomials, exponential function e xand the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square rootthe logarithmthe trigonometric function tangent, and its inverse, arctan.

For these functions the Taylor series do not converge if x is far from b.

1+ x)^- 1/ 2 taylor expansion

That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. The pink curve is a polynomial of degree seven:. The error incurred in approximating a function by its n th-degree Taylor polynomial is called the remainder or residual and is denoted by the function R n x.

Taylor's theorem can be used to obtain a bound on the size of the remainder. In general, Taylor series need not be convergent at all.

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For example, the function. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functionswhich might have singularities, never converge to a value different from the function itself. More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma.Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. I know that we can differentiate the function and substitute in the Taylor formulabut I want a shorter way. You have already done what you want, I guess. Your question was kind of unclear to me at first because you said you didn't want to do any substitutions at all, so let me see if I can make my point understood by expressing it in detail.

Taylor series

Suppose that we're dealing with formal powers series and ignore analytic and convergence issues for now. Therefore, as a bonus, you get:. This is precisely what you need to do to your series and any method of calculating what you want is something like this in diguise. Now if your function happens to be analytic in a domain and you stay at a point in that domain, you're fine but your radius of convergence is certainly changed. Sign up to join this community.

The best answers are voted up and rise to the top. Asked 3 years, 2 months ago. Active 3 years, 2 months ago. Viewed 28k times. Michael Hardy k 28 28 gold badges silver badges bronze badges. I am sorryI did not concentrate while writing it!

Add a comment. Active Oldest Votes. But I don't see how any method can show what the OP wants. The OP says they want to do it without substitution. I'm not sure about what the OP wants.The website and information I was sent gave me a good idea of what my holiday was going to be like.

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Taylor Series Calculator with Steps

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1+ x)^- 1/ 2 taylor expansion

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1+ x)^- 1/ 2 taylor expansion

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