What is the general procedure for using Taylor Series to evaluate:
i) sums
eg.\sum_{n=4}^{\infty }\frac{n(n-1)2^n}{3^n}
ii) limits
eg. \lim_{x\rightarrow 2}\frac{x^2-4}{ln(x-1)}
iii) derivatives
eg. Find f^{(11)}(0) of f(x)=x^3sin(x^2)
iv) integrals
eg. \int_{0}^{1} \frac{1}{2-x^3}dx
I was asked to find sums equal to 9/25 by using the power series of y=\frac{1}{1+x^2}. First thing I did was to find the power series representation of the function:
\sum_{n=0}^{\infty }(-x^2)^n
Next I figured out the interval of convergence:
\left \| -x^2 \right \|< 1
This meant that x...
How would I find the interval of convergence for the following series:
i) \sum \frac{(x+2)^n}{n^2}
ii) \sum \frac{(-1)^kk^3}{3^k}(x-1)^{k+1}
iii) \sum (1+\frac{1}{n})^nx^n
Reason for edit: My second series was not displaying properly