Use the Divergence Test to determine whether the following series diverges or state that the test is inconclusive. Summation from k equals 0 to infinity StartFraction 1 Over 1000 plus k EndFraction

Answer:The test is inconclusive.Step-by-step explanation:Let us assume that the series is [tex]\sum_{k=0}^\infty \frac{1}{1000+k}[/tex]. The Divergence Test tell us that, if the limit of the sequence defined by the general term of the series: [tex]\lim_{k\rightarrow \infty}a_k[/tex], is different from zero, or it doesn't exist, then the series diverges.In this exercise we have [tex]a_k= \frac{1}{1000+k}[/tex], then the limit we want to study is[tex]\lim_{k\rightarrow \infty} \frac{1}{1000+k}.[/tex]It is not difficult to see that if [tex]k[/tex] grows to infinity,the limit of the given fraction is zero:[tex]\lim_{k\rightarrow \infty} \frac{1}{1000+k}=0.[/tex]Thus, the convergence test is inconclusive. Recall the case of the harmonic series: [tex]\sum_{k=1}^\infty \frac{1}{k}[/tex], which is divergent and clearly [tex]\lim_{k\rightarrow \infty}\frac{1}{k}=0[/tex].