(Solved): 1. For this discussion, find an example of a wrong comparison test. That is, when comparing the se ...

1. For this discussion, find an example of a wrong comparison test. That is, when comparing the series to a simpler series, the inequality is not in the right direction (refer to the Direct Comparison Test and related comments in the notes). Once such example is the series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}+1}} \). When comparing this series with a simpler series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}}}=\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}}}=\sum_{n=1}^{\infty} \frac{1}{n} \), we see that since \( \frac{1}{\sqrt{n^{2}+1}}<\frac{1}{n} \) and the (larger) harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges, we can't say anything about the smaller series (that's why the inequality \( < \) is not useful in this case, it is not in the right direction). 2. Clearly show why the inequality is not useful when using the Direct Comparison Test (Do not use the example above or any similar examples in the notes). As always, you must embed an image ?_of your work for credit (uploading a pdf in any discussion just won't work. Also, don't just upload an image as an attachment, embed it!) 3. Use the Direct or Limit Comparison Test to determine the convergence of the series in your example.