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