Use an appropriate test to determine whether the following series converges.

Ask by Conner Williams. in the United States

Mar 30,2025

Question

Use an appropriate test to determine whether the following series converges. $$ \sum_{k=6}^{\infty} \frac{1}{(k-5)^{5}} $$

UpStudy AI · Accepted Answer

1. Let $$ n = k - 5 $$. Then the series becomes
   $$
   \sum_{k=6}^{\infty} \frac{1}{(k-5)^{5}} = \sum_{n=1}^{\infty} \frac{1}{n^5}.
   $$

2. The series $$\sum_{n=1}^{\infty} \frac{1}{n^5}$$ is a $$ p $$-series with $$ p = 5 $$.

3. Recall that a $$ p $$-series $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ converges if $$ p > 1 $$.

4. Since $$ 5 > 1 $$, the series converges by the $$ p $$-series test.

Thus, the series converges.
The series $$ \sum_{k=6}^{\infty} \frac{1}{(k-5)^{5}} $$ converges because it is a $$ p $$-series with $$ p = 5 $$, and since $$ p > 1 $$, the series converges.

Use an appropriate test to determine whether the following series converges.

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