Problem 5 - Calculus Early Transcendentals

Question

For what values of \(p\) does the series \(\sum_{k=10}^{\infty} \frac{1}{k^{p}}\) converge (initial index is 10 )? For what values of \(p\) does it diverge?

Step-by-Step Solution

Answer

Answer: The series converges for values of \(p > 1\) and diverges for values of \(p \leq 1\).

Step 1: Understand the series type

The given series \(\sum_{k=10}^{\infty} \frac{1}{k^{p}}\) is a type of p-series, which is in the form of \(\sum_{k=1}^{\infty} \frac{1}{k^{p}}\). In this case, the initial index is given as 10, but this does not affect the convergence or divergence of the series.

Step 2: Apply the p-series test

The p-series test states that the series \(\sum_{k=1}^{\infty} \frac{1}{k^{p}}\) converges if \(p > 1\) and diverges if \(p \leq 1\). In this case, we have the series \(\sum_{k=10}^{\infty} \frac{1}{k^{p}}\). The same rule applies due to having the same general form.

Step 3: Determine the values of \(p\) for convergence and divergence

The series \(\sum_{k=10}^{\infty} \frac{1}{k^{p}}\) converges if \(p > 1\). Thus, for values of \(p\) where \(p > 1\), the series converges. On the other hand, the series diverges if \(p \leq 1\). That means for values of \(p\) where \(p \leq 1\), the series diverges.

Step 4: Conclusion

The series \(\sum_{k=10}^{\infty} \frac{1}{k^{p}}\) converges for values of \(p > 1\) and diverges for values of \(p \leq 1\).