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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

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Answer:

The test is inconclusive.

Explanation:

Let us assume that the series is
\sum_(k=0)^\infty (1)/(1000+k). The Divergence Test tell us that, if the limit of the sequence defined by the general term of the series:
\lim_(k\rightarrow \infty)a_k, is different from zero, or it doesn't exist, then the series diverges.

In this exercise we have
a_k= (1)/(1000+k), then the limit we want to study is


\lim_(k\rightarrow \infty) (1)/(1000+k).

It is not difficult to see that if
k grows to infinity,the limit of the given fraction is zero:


\lim_(k\rightarrow \infty) (1)/(1000+k)=0.

Thus, the convergence test is inconclusive. Recall the case of the harmonic series:
\sum_(k=1)^\infty (1)/(k), which is divergent and clearly
\lim_(k\rightarrow \infty)(1)/(k)=0.

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