# expansion

1. ### Maclaurin expansion, and third derivative

Hi, I'm having some trouble with the following problem: Let f(x)be a real valued, continuous function, differential on (-k,k) for k \in \mathbb{R} . The graph of f(x) is a u shape, with vertical asymptotes at -k, and +k. The base of the u is at x=0. let the Maclaurin expansion of...
2. ### Taylor Series Expansion?

Hi. I'm reading this one book about pulsatile flow and there's one passage in it that me and my friends can't make sense of (Headbang). You don't need much insight into the physical problem to understand it, so I figured I'd ask for help here. Also, the author probably used a Taylor series...
3. ### Power Series Expansion

Obtain the power series expansion of \frac { 3 }{ (1+x)(1-2x) } , giving the general term and radius of convergence.
4. ### Binomial?

Hi all, I'm new here at MHF and i really need help with this binomial homework Given that the coefficients of x and x2 in the expansion of (1+ax)6(2+bx)5 are -112 and 80 respectively, find the integer values of a and b
5. ### Reevaluating an Indefinite Integral

The indefinite integral of (ln(x^e^x))^2 Wolfram Alpha will give me the answer but it will not give a step by step explanation, that would so much appreciated. My calculator gives a another form. The form I'm interested in first of all is the one that Wolfram gives, involving the Lamda...
6. ### Finding limits using power series expansion

Hi, I have this question: Using a power series expansion, and not otherwise, find \lim_{x \to 0} (\frac{1}{1-cosx}-\frac{2}{x^2}) I have expanded the cosx with the power series, but then don't understand how to get to the answer from that (ans: \frac{1}{6}) Many thanks for the help in advance.
7. ### Unitary transformation in commutator expansion

I would like a proof or an outline of one for the following identity: exp(A)*B*exp(-A) = B + [A,B] / 1! + [A,[A,B]] / 2! + ... + [A,[A,...,[A,B]...] / n! + .... A and B are linear operators in the case I am considering, but the identity is a formal one, which I believe depends only on the...
8. ### limit/power series expansion

I uploaded an example from my probability book that uses limits. Can someone show me the steps on how it goes from #1 to #2? #1 log M_{Z_{n}}(t) = - t\sqrt{\lambda_{n}} + \lambda_{n}(e^{\frac{t}{\lambda_{n}}}-1) #2 \lim_{n\to\infty} M_{Z_{n}}(t) = \frac{t^2}{2}
9. ### Expansion

How many dissimilar terms are there in the expansion (2x-3y+5z)^20?
10. ### expansion

Help to expand (1-e-a)b Thanks Seyi
11. ### Binomial Expansion

Expanding (2+h)^3 I get 8h+8h^2+2h^3 but... the answer is 8+12h+6h^2+h^3 ...What am i doing wrong?
12. ### Determinant (without expansion)

Can anyone help me with this (Worried) Evaluate using the properties, x+y y+z z+x z x y 2 2 2 for example - Column 1 tends to Column 1 - Column 2, Row 3 tends to Row 3 - 2 X Row 1...
13. ### binomial expansion

find the possible values of a and b if the expansion in ascending powers of x up to the term in x^2 of sqrt(1+ax)/(1+bx)=(1-9/2)x^2 my steps is as below first i expand (1+ax)^1/2=1+1/2ax-1/8a^2 x^2--eq 1 (1+bx)^-1=1-bx+2b^2x^2--eq 2 eq1/eq2,1-(1/8a^2)/(2b^2)x^2=9/2 then by compare...
14. ### Binomial Expansion

Hi all. My problem is: What is the coefficient for x^3 in (2x+4)^8. So I know that I am dealing with the binomial theorem. So I started with the exponents only using the formula a^(n-k)*b^k. This is what I have: K=0 a^8*1 K=1 a^7*b k=2 a^6*b^2 k=3 a^5*b^3. The answer that the webpage is...
15. ### Expansion of a functional

Hi I'm given the functional S\left[y\right]=\int_{1}^{2}\sqrt{x^{2}+y^{\prime}}dx\qquad y(1)=0,\: y(2)=B and asked to find \Delta=S[y+\epsilon g]-S[y] to the second order in \epsilon, so with a Taylor expansion I get...
16. ### Find matrix elements and expansion for operator in position representation

In 1-D let TL be an operator defined on the position eigenstates |x> such that TL|x>=|x+L>. Find the matrix elements TL(x,x')=<x'|TL|x> and construct an explicit expansion for this operator in the position representation. Show that “in the position representation” <x|T_L|\psi >= \psi (x-L)...
17. ### Fourier expansion not admissible in dispersive multi-mode wave propagation ?

Dear Group, I kindly ask your expertise. We have a signal of strain versus time which I expand into a Fourier series. The attached graph shows on the left the recorded signal in red and the Fourier expansion in four different versions (sum of a_n*Sin(n*f*t) + b_n*Cos(n*f*t) and sum of...
18. ### Taylor expansion??

Hi, me again. I have a question as follows: "Obtain the first four terms of the expansion x around t=1 if x(1) is real and x^3 + 2xt -3 = 0 " I have no idea where to start, please can you help me. I have no proper example in my available text books :( Thank you ever so much Kas
19. ### Binomial Expansion theory and series

Hello, We know the following: (1+x)^k = \sum_{n=0}^{\infty} {k\choose {n}} \cdot x^n. So from f(x) , k = 1/4 , and we plug in (-x) in for x. But then I am having trouble calculating what {k\choose n } is equal to. In my notes the prof wrote down: {k\choose n } = \frac{k!}{(k-n)}!n! but...
20. ### Taylor expansion

Hi, what is the simplest way of proving e^{ix}=\cos x + i \sin x=\sum\limits_{k=0}^{n-1}\frac{(ix)^k}{k!}+\frac{(ix)^n}{n!}(\cos{\theta_2x}+i\sin{\theta_1 x}) for \theta_1,\theta_2 \in (0,1) ??? I would just do the taylor expansion of \cos x and of \sin x separately. Thanks in advance!