X N X X X N
So, the series converges when $|x| Product of exponentials with same base. If we take the product of two exponentials with the same base, we simply add the exponents:
$$f(x) = \sum_{n=1}^\infty n. Webthe power rule is used to differentiate the algebraic expressions of the form x^n. Webfirst remark that the polynomial x^m+k has simple roots.
For the function f (x) = xn, n should not equal 0, for reasons which will become clear. Xaxb = xa + b. Webwhat is the derivative of xn? Let us call the roots c_j for 1\leq j\leq m. Therefore x^n/(x^m+k) rewrites as a sum of simple elements.
Webusing the ratio test, $\dfrac{a_{n+1}}{a_n} = \dfrac{(n+1)x^{n+1}}{nx^n} = \dfrac{(n+1)x}{n}$.
