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Let \( P_{k} \) be a point in \( x y \) plane whose \( x \) coordinate is \( 1+\frac{k}{n}(k=1,2,3, \ldots \ldots ., n) \) on the
P curve \( y=\ln x \). If A is \( (1,0) \), then \( \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(A P_{k}\right)^{2} \) equals :

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(a) \( \frac{1}{3}+2 \ln ^{2} 2 \)
(b) \( \frac{1}{3}+2 \ln ^{2}\left(\frac{2}{e}\right) \)
(c) \( \frac{1}{3}+\ln ^{2}\left(\frac{2}{e}\right) \)
(d) \( \frac{1}{3}+2 \ln \left(\frac{2}{e}\right) \)
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