# expansions

1. ### Power Series Expansions

Hi, How do I get the power series expansion for e^x ln(1+x)? Subbing 0 into that gives 0. If I differentiate that I get e^x ln(1+x) + e^x 1/x + x. Subbing 0 into that gives 0 again, the correct answer is 1/2... Where have I gone wrong? Help is much appreciated.
2. ### Binomial expansions

By writing the terms of the expansions of (2x+1)^3 in ascending powers of x , find (3+2x^2)^3 For the expansion of (2x+1)^3 = 1 +6x + 12x^2 + 8x^3 But i am confuse at the second part of the question. Any help ? Thank you
3. ### 2-adic expansions of inverse powers of 3

I am trying to teach myself about p-adic numbers, and I came across an interesting property. Consider the following expansions: \displaystyle \begin{align*}-\frac{1}{3} & = \ldots 010101._2 \\ -\frac{2}{3} & = \ldots 101010._2 \\ -\frac{1}{9} & = \ldots 000111._2 \\ -\frac{2}{9} & = \ldots...
4. ### Coefficient and expansions.

find the Coefficient of term x2y3 in below expansion: expansion : (2+x-y+3z)10 please write a good description for solution* or how to solve it? * for a person who is a dummy in math and combinatorics...
5. ### Stratergies for tricky Taylor Expansions

I am required to taylor expand ln(sec(x)) to obtain the first 3 non-zero terms. What I wanted to know was if anyone knows a quicker method than differentating 6 times (this gets very messy)? Thanks in advance.
6. ### Laurent Expansions

I am stuck at two questions about Laurent expansion. Below are the problems: 1. Determine the Laurent expansion of (z^2 + 2z)/(z^3 -1) in the annulus 0 < | z - 1 | < R about z = 1. Find the large R one can use. 2. Obtain a Laurent expansions of f(z) =1/[ (z-j) (z-2) ] in the region 0 <...
7. ### Help with Laurent Expansions

Basically I've been working through the problem below, and when I went to check my answers they differed, so I was wondering are the answers below the question both the same? Thanks.
8. ### Help with binomial expansions

Is there a quick way to expand: (3x + 5)(x - 7)? (Crying)
9. ### Checking Taylor Expansions

Hi, I just want to check that I have the Taylor expansions correct for these choices. I cannot find any examples on the net. So would be great if you could give me these. y(x_{i+1}) y(x_{i+2}) y'(x_{i-2}) y'(x_{i+1}) Thanks in advance
10. ### Binomial expansions

Find, in the simplest form, the coefficient of x^n in the binomial expansion of (1-x)^(-6). Hence, find the coefficient of x^6 and x^7 in (1+2x+3x^2+4x^3+5x^4+6x^5+7x^6)^3 Attempt: The generalised binomial theorem, \sum^{\infty}_{k=0}\frac{-6(-6-1)(-6-2)...(-6-r+1)}{r!}(1)^{-6-r}(-x)^r So...
11. ### Binomial Expansions, what is the term

Hello! This is the question, find the term x^3 in (3x +4 ) (x − 2)^4 I dont know how to start or what is ment by the 'term', do i expand (x − 2)^4? Thank you!!(Nod)(Nod)(Nod)
12. ### mixed bag part 1:assessing parabolas, and binomial expansions

Some mixed questions I need some confirmation on. 1: Find (a) the directrix, (b) the focus and (c) the roots of the parabola y=x^2-5x+4. I can find the directrix and the focus no problem, but i do not understand what they mean by "roots". Do they mean the roots(positive and negative) of the...
13. ### Binomial Expansions

Q1)Find, in its simplest form, the coefficient of x^r in the expansion, in ascending powers of x , of \frac{1}{x-3} Q2) Expand (1+y)^14 as a series of ascending powers of y up to and including the term in y^3, Simplify the coefficients. I got the above part right and derived at the...
14. ### Expansions

f(x)=\frac{2x}{(x+1)^2(x^2+1)}=\frac{1}{x^2+1}-\frac{1}{(x+1)^2} Find the seires expansion of \log \left[ \frac{f(x)}{2x}\right] up to [x^4] \log\left[\frac{f(x)}{2x}\right]=\log\left[\frac{1}{(x+1)^2(x^2+1)}\right] I suppose then we would have -2\log(x+1)-\log(x^2+1) but i don't know...
15. ### Laurent expansions

Find all Laurent expansions centered at z0 = 0 for f(z) = 1/[z^2(z^3 + 1)] and find the region of convergence for each expansion. For this problem, I start with rewrite f(z) = 1/(z^2) *1/ [1-(-z^3)] then use geometric series to write laurent series representation for 1/ [1-(-z^3)]. I am stuck...
16. ### Laurent expansions

In my textbook, as part of an example, I am given that "\frac{1}{{{{\left( {z - 1} \right)}^2}}} is its own Laurent expansion about z = 1, where it has a double pole." My reasoning is this: If we define a punctured disc centred at 1 with radius r > 0, then we can express it as a Laurent series...
17. ### Binomial Expansions with negative and fractional Expontents

I know, I know two threads in 5 minutes, but I can't get this one! Binomial expansions are all fine and dandy, but when the exponent is a negative, a fraction, or both I am a little stumped. Write the fourth term in the binomial expansion for: (a - b)^-(1/5) and I don't even know where to begin!
18. ### Ternary Expansions and Cantor Set

The problem is to find out for what values of p (for integers between 0 and 13), is p/13 in the cantor set. I know that they are only in the cantor set if they can be expressed in the ternary expansion to base 3 using only 2's and 0's. I have a lot of problems with these ternary expansions so...
19. ### Series expansions

I'm attempting this question on binomial distributions: 1 + mx + \frac{m(m-1)x^2}{2} are the first 3 terms of the series expansion of (1+x)^m Use this to find the 1st 3 terms of the series expansion of (1+x)^{m+1} (1-2x)^m This is what I've done: 1) Substituted (m+1) into the initial...
20. ### Rational functions and approximate expansions

Hi, I know how to do normal binomial expansions but I can't remember questions quite like this which I've just found in my textbook, by exam being in just over a day. The first bit is to show that (1+x)/(2+x) - (1-x)/(2-x) = 2x/(4-x^2). I did that fine. The second bit says 'Hence or otherwise...