Determine if the following series converges or diverges. If it is a converging geometric or telescoping series, or ca be written as one, provide what the series converges to. ∑k=2∞kk3−17

Question

Determine if the following series converges or diverges. If it is a converging geometric or telescoping series, or ca be written as one, provide what the series converges to.  ∑k=2∞kk3−17

Maciej Morrow · Accepted Answer

The series is ∑k=2∞kk3−17Limit Comparison test:  Suppose ∑an and ∑bn are two series and limn→∞anbn=c. If c is a finite and positive number then both the series converge or diverge together.Let&#039;s take the second series ∑bn=∑1n2 which is known as convergent by p-series.∑an=∑nn3−17∑bn=∑1n2limn→∞anbn=limn→∞nn3−171n2=limn→∞nn3(1−17n3)×n21=limn→∞1(1−17n3)=1As the ration of the two series is 1 which is finite and positive, using Limit Comparison test the series ∑an=∑nn3−17 converges as the series ∑bn=∑1n2 convergent.Hence ∑k=2∞kk3−17 is convergent.

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Determine if the following series converges or diverges. If it is a converging geometric or telescoping series, or ca be written as one, provide what the series converges to. sum_{k=2}^inftyfrac{k}{k^3-17}

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New Questions in Calculus 2