bobcantor1983
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- Joined
- Oct 1, 2013
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Let \(\displaystyle x_n = 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} - \ln(n)\) where \(\displaystyle n = 1,2,3,...\)
Prove that the sequence \(\displaystyle {x_n}\) converges.
Hint: Show, first, that \(\displaystyle x > \ln(1+x)\) for any \(\displaystyle x > 0\). Then show that the sequence \(\displaystyle {x_n}\) increases and \(\displaystyle x_{n+1} - x_n < \frac{1}{2(n+1)^2}\). Use the theorem about the convergence and divergence of p-series to complete the proof.
I don't really understand how these hints relate to the proof or how to start. Thoughts?
Prove that the sequence \(\displaystyle {x_n}\) converges.
Hint: Show, first, that \(\displaystyle x > \ln(1+x)\) for any \(\displaystyle x > 0\). Then show that the sequence \(\displaystyle {x_n}\) increases and \(\displaystyle x_{n+1} - x_n < \frac{1}{2(n+1)^2}\). Use the theorem about the convergence and divergence of p-series to complete the proof.
I don't really understand how these hints relate to the proof or how to start. Thoughts?