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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