Question 8.3.6: Using the Comparison Test for a Divergent Series Investigate...

From the graph of the first 20 partial sums seen in Figure 8.27, it appears
that the partial sums are growing very rapidly. On this basis, we would conjecture that the series diverges. Of course, to verify this, we need further testing. Notice that for k large, the general term looks like \frac{5^k}{2^k}=\left(\frac{5}{2}\right)^k and we know that \sum_{k=1}^{\infty}\left(\frac{5}{2}\right)^k is a divergent geometric series \left(|r|=\frac{5}{2}>1\right) . Further,

By the Comparison Test, \sum_{k=1}^{\infty} \frac{5^k+1}{2^k-1}. diverges, too.