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Chapter 9: Problem 29

Determine the convergence or divergence of the following series. $$\sum_{k=1}^{\infty} \frac{1}{k^{10}}$$

Short Answer

Expert verified
Explain. Answer: The series converges. This is because the series is a p-series with p = 10, which is greater than 1. According to the p-series test, a p-series converges if the exponent p is greater than 1, as the terms get smaller and approach zero, leading to a finite sum.

Step by step solution

01

Determine p value in p-series

The series given is a p-series, with the general formula: $$\sum_{k=1}^{\infty} \frac{1}{k^p}$$ For our particular series, we have: $$\sum_{k=1}^{\infty} \frac{1}{k^{10}}$$ The value of p in this case is 10.
02

Compare p to 1

We need to compare the value of p to 1 in order to determine the convergence or divergence of the series. In our case, p = 10. Comparing 10 to 1: $$10 > 1$$
03

Determine convergence or divergence using p-series test

Since the value of p (10) is greater than 1, the p-series test tells us that the given series converges: $$\sum_{k=1}^{\infty} \frac{1}{k^{10}}$$ converges. The series converges because the given p-series has an exponent greater than 1, which means that the terms get smaller and smaller as the series goes on, ultimately getting closer and closer to zero. This also means that the sum of the series approaches a finite value as more terms are added.

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