Does S converge? (The answer says it converges)

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> $S=\frac{1}{\ln 2} \sum\_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$
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My attempt:

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> Comparison test: $\ln (1+\frac{1}{k(k+2)}) \ln k \lt \ln 2 \ln k$ --- not work;
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> Cauchy's test: $\lim\_{k \to \infty} |\ln (1+\frac{1}{k(k+2)}) \ln k|^{1/k} \lt 1$? --- hard to tell
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> Ratio test: $\lim\_{k \to \infty} |\frac{\ln (1+\frac{1}{(k+1)(k+3)})\ln (k+1)}{\ln (1+\frac{1}{k(k+2)})\ln k} | \lt 1$? --- hard to tell
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> Integral test: is the series decreasing and founded?
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Thanks,