# Series Integration

Calculus Level 5

$D(x) = \int (\sum_{n=1}^{\infty} nx^{n-1})dx = \int S_ndx\\ and \\ J(x) = D(x) - c \\ for \hspace{1mm} |x| < 1$ Determine: $$J(0)$$.

$$Note:$$ In this case, $$c$$ is the arbitrary constant from integration.

The following is a well known converging $$geometric$$ $$series$$ that may prove useful: $\sum_{n=1}^{\infty} x^n = \frac{1}{1-x}$

$$Hint:$$ First determine what the series converges (or diverges) to; i.e, determine $$S_n$$. Then determine $$\int S_n dx$$; this antiderivative will equal $$D(x)$$. Perform these two steps before evaluating $$J(x)$$ at $$x=0$$.

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