Taylor Series Expansion of Log(1+x)

This power point highlights the way of solving log(1+x) using Taylor's expansion. Also there are brief discussion about the Formula and certain examples of other such series. The slides are good enough for an engineering term paper of mathematics.

1. Estimate of Infinite Series for the function Log(1+x) around the points x=5 & x=10 Represented By : Arijit Dhali

2. Abstract In this presentation , our aim is to solve the theory using Taylor series. Throughout this presentation we will gather the basic knowledge about the topic . This presentation is a team work and every member is associated with this presentation has collected the data from various trusted sources.

3. Keywords Continuity Derivative Taylor’s Series Lagrange form of remainder

4. Introduction In calculus,  Taylor's theorem gives an approximation of a  k -times differentiable function around a given point by a polynomial of degree  k , called the  k th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order  k  of the Taylor Series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician  Brook Taylor , who stated a version of it in 1715 , although an earlier version of the result was already mentioned in  1671 by  James Gregory . A Maclaurin series is a function that has expansion series that gives the sum of derivatives of that function. The Maclaurin series of a function f(x) up to order n may be found using Series [f,x,0,n]. 

5. Discussion Continuity: Continuity is the formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. Derivative: Derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Taylor Series: 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. Lagrange Remainder: Lagrange remainder is: where M is the maximum of the absolute value of the (n + 1) th derivative of function on the interval from x to c. The error is bounded by this remainder.

6. Taylor’s Theory Let f(x) : [a,b] -> R, be a function such that , (n-1) th derivative of the function f(x) is continuous in [a,b] The nth derivative of the function f(x) exist in (a,b) Then , f(b) = where Rn is known as the Remainder in Lagrange’s Alternate Form, If , Rn = Then , it is known as the Taylor Series Expansion in Lagrange’s Alternate Form.  

7. Taylor Series of Other Functions

8. Example Example : Write a Taylor’s formula for the function , f(x) = log(1+x), -1<x<∞ about x=2 with Lagrange form of remainder after 3 terms. Solution : Here the function f(x)=log(1+x), -1<x<∞ is continuous and derivable at each point of (-1, ∞). Now, f’(x)= , f’’(x)= , f’’= …..and so on. Now the Taylor’s formula for the function f(x) about x=2 is, Since, f (2) = log3, f’ (2) = 1/3, f’’ (2) = -1/9 Therefore, Log (1+x)  

9. Calculation: 1 Infinite Series for the function Log(1+x) around x=5 Here the function f(x) = log(1+x), -1<x<∞ is continuous and derivable at each point of (-1, ∞) Now, f(x) = log(1-x) f’(x) = f’’(x) = f’’’(x) = and so on……..   Now using Taylor’s formula for the function f(x) about x=5 is, f(x) = f(5) + (x-5) f’(5) + f’’(5) + f’’’(5) + …+ , 0 < θ < 1   Since, f(5) = log(6), f’(5) = , f’’(5) = , f’’’(5) = …………..and so on. Therefore, f(x) = log 6 + (x-5) + ( ) + ( ) +……+ Log(11) = 1 + (x-10) - + -……+  

10. Calculation: 2 Infinite Series for the function Log(1+x) around x=10 Here the function f(x) = log(1+x), -1<x<∞ is continuous and derivable at each point of (-1, ∞) Now, f(x) = log(1-x) f’(x) = f’’(x) = f’’’(x) = and so on……..   Now using Taylor’s formula for the function f(x) about x=5 is, f(x) = f(10) + (x-10) f’(10) + f’’(10) + f’’’(10) + …+ , 0 < θ < 1   Since, f(10) = log(10), f’(10) = , f’’(10) = , f’’’(10) = …………..and so on. Therefore, f(x) = 1 + (x-10) + ( ) + ( ) +……+ Log(11) = 1 + (x-10) - + -……+  

11. Conclusion This project has been beneficial for us , as it enabled us to gain a lot of knowledge about the use of Taylor Series about a appoint , as well as its applications. It also helped us to develop a better coordination among us as we shared different perspectives and ideas regarding the sub-topics we organized in this project .

12. References https://en.wikipedia.org/wiki/Taylor_series https://math.stackexchange.com https://www.quora.com/What-is-the-expansion-of-log-1-x Engineering Mathematics I – B.K.Pal & K.Das