X N X X X N - Poursteady Utilities

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.

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.

Webwhat is the derivative of xn?

For the function f (x) = xn, n should not equal 0, for reasons which will become clear.

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Product of exponentials with same base.

If we take the product of two exponentials with the same base, we simply add the exponents:

Xaxb = xa + b.

Webusing the ratio test, $\dfrac{a_{n+1}}{a_n} = \dfrac{(n+1)x^{n+1}}{nx^n} = \dfrac{(n+1)x}{n}$.

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So, the series converges when $|x|

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$$f(x) = \sum_{n=1}^\infty n.