







Infinite Logarithmic Series 
20110808 

From Sourik: Dear Expert,
In my Amithabha Mitra and Shambhunath Ganguly's "A Text Book of Mathematics" I found the formula of log (1+x) where the base is e and x lies in between 1 and +1.As I want to learn Mathematics,I am not satisfied with the mere statement of the formula.Please help giving me the full proof.
Thanking you,
Sourik Answered by Robert Dawson. 





A Taylor polynomial for (lnx)/x 
20100929 

From Dave: I have a series problem that I cannot solve. The problem asks for you to compute a Taylor polynomial Tn(x) for f(x) = (lnx)/x. I calculated this poly out to T5(x) and attempted to use this to identify a pattern and create a series in order to calculate Tn(x). However, the coefficients on the numerator out to F5prime(x) are as follows: 1, 3, 11, 50, 274... Ok, so the negative is an easy fix > (1)^n1. But the other coefficients are stumping me. I can't see any sort of pattern there and I've tried every trick I know. Is there another way to go about this?
Thanks! Answered by Chris Fisher. 





The Maclaurin series generated by f(x)=x^ cosx + 1 
20050810 

From Latto: f(x)=x^{3}·cosx + 1. but when I take the derivatives, I couldn't see a pattern. Can you help?
Answered by Penny Nom. 





A Taylor series for ln(x) 
20050416 

From Anood: i have to represent ln(x) as a power series about 2
i`m not getting the final answer which is ln 2+ sigma (((1)^{(n+1)}/
(n*2^{n}))*(x2)^{n}). i don`t get the ln 2 part
i show you my trial
f(x)= ln x.
f(x)=(1/x) .
f(x)= (1/x^{2})*1/2!
f(x)= (2/x^{3})*1/3!
f(x)= (6/x^{4})* 1/4!
so the pattern shows me that f(n)= ((1)^{(n+1)})/x^{n} *n)
so f(2)= sigma ((1)^{(n+1)})/2^{n} *n) *(x2)^{n}
so as you see i don`t get ln 2
Answered by Penny Nom. 





The third derivative 
20041015 

From Holly: Why would you ever take the third derivative? Answered by Harley Weston. 





Programming without trig functions 
20040525 

From Derek: I am a programmer trying to calculate the following.
What is the formula to find the crosssectional area of a cylinder with out using any trig functions? or better yet, how can you calculate any given volume in a cylindrical tank with spherical heads with out trig functions?
I am using a PLC (programmable logic controller) to do this and trig functions are not available. Answered by Harley Weston. 





Cosine of 35 degrees 
20040303 

From Jason: How do you find the exact solution to cosine 35 degrees. Answered by Chris Fisher. 





A Taylor series 
20010427 

From Karan: Given the following information of the function  f''(x) = 2f(x) for every value of x
 f(0) = 1
 f(0) = 0
what is the complete Taylor series for f(x) at a = 0 Answered by Harley Weston. 





Maclaurin series again 
20000923 

From Jason Rasmussen: I suppose my confusion comes into play when I am trying to figure out where the x^{n} term comes from. I know that the Power Series notation is directly related to the Geometric Series of the form sigma [ br^{n} ] where the limit is b/(1r) for convergence at  r  <1. Therefore, the function f(x) needs to somehow take the form of b/(1(xa)), which may take some manipulation, and by setting r = (xa) and b = C_{n}, we get the Geometric Series converted to the Power Series. Taking the nth order derivative of the Power Series gives C_{n} = f^{n}(a)/n!. There must be a gap in my knowledge somewhere because I cannot seem to make f(x) = e^{x} take the form of f(x) = b/(1(xa)). Maybe I should have labelled my question as "middle" because it may be more of a personal problem with algebra and logarithms. Or, am I to assume that all functions can be represented by sigma [f^{n}(a) * (xa)^{n} / n!]? Answered by Harley Weston. 

