1. ## Maclaurin Series questions

I need help with various calculus review problems. that would be great if you could.

E) What are the first four terms of the MacClaurin series of $\displaystyle f(x) = \frac {1}{1 = 2x}$? What is its interval of convergence?

E1) What are the first four terms of the MacClaurin series of $\displaystyle f(x) = \frac {1}{1 = 2x}^2$? What is its interval of convergence?

E2) What are the first four terms of the MacClaurin series of $\displaystyle ln(1 + 2x)$? What is its interval of convergence?

2. Originally Posted by corgonin
I need help with various calculus review problems. that would be great if you could.

E) What are the first four terms of the MacClaurin series of $\displaystyle f(x) = \frac {1}{1 = 2x}$? What is its interval of convergence?

E1) What are the first four terms of the MacClaurin series of $\displaystyle f(x) = \frac {1}{1 = 2x}^2$? What is its interval of convergence?

E2) What are the first four terms of the MacClaurin series of $\displaystyle ln(1 + 2x)$? What is its interval of convergence?
... and we apply the binomial expansion...

$\displaystyle (1+a\cdot x)^{n}= 1 + n\cdot (ax) + \frac{n\cdot (n-1)}{2}\cdot (ax)^{2} + \dots + \frac{n\cdot (n-1) \dots (n-k+1)}{k!}\cdot (ax)^{k} + \dots$ (1)

We obtain...

$\displaystyle \frac{1}{1+2x} = 1 - 2\cdot x + 4\cdot x^{2} - \dots + (-1)^{k}\cdot (2x)^{k} + \dots$ (2)

$\displaystyle \frac{1}{(1+2x)^{2}} = 1 - 4\cdot x + 12\cdot x^{2} - \dots + (-1)^{k}\cdot (k+1)\cdot (2x)^{k} + \dots$ (3)

Because is...

$\displaystyle \ln (1+2x)= 2\cdot \int \frac{dx}{1+2x}$

... from (2) ...

$\displaystyle \ln (1+2x) = 2\cdot x -2\cdot x^{2} + \frac{8}{3}\cdot x^{3} + \dots + (-1)^{k}\cdot \frac{2^{k+1}}{k+1}\cdot x^{k+1} + \dots$ (4)

The expansion (2), (3) and (4) converge all for $\displaystyle |x|<\frac{1}{2}$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$