• A. According to the Divergence Test, the series diverges because lim ax

(Simplify your answer.)

O B. According to the Divergence Test, the series converges

because lim ax

k->00

(Simplify your answer.)

O C. The Divergence Test is inconclusive because lim ax =

Question

College

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Answer :

mboluwape mboluwape · Answer 1 · 2024-12-22T18:45:37-05:00

The limit of the general term is 0, so we can conclude that the Divergence Test is inconclusive for the given series. C

To apply the Divergence Test to the given series:

Σ [tex]K^ 3/ (7k^3 + 1)[/tex]

k = 0

Evaluate the limit of the general term of the series as k approaches infinity.

Taking the limit of the general term:

lim ([tex]K^3/ (7k^3 + 1[/tex])) as k approaches infinity

To simplify the limit, divide both the numerator and denominator by [tex]k^3[/tex]:

lim[tex](K^3/ (7k^3 + 1)) = lim (K^3/k^3) / (7 + 1/k^3)[/tex] as k approaches infinity

Now, as k approaches infinity, 1/[tex]k^3[/tex] approaches 0:

[tex]lim (K^3/ (7k^3 + 1)) = lim (K^3/k^3) / (7 + 1/k^3)\\= lim (K^3/k^3) / 7[/tex]

Since the numerator is a power of k and the denominator is a constant, the limit becomes:

lim ([tex]K^3/ (7k^3 + 1[/tex])) = 0/7 = 0 as k approaches infinity

The limit of the general term of the series is 0.

According to the Divergence Test, if the limit of the general term is not equal to zero, the series diverges.

However, if the limit is equal to zero, the test is inconclusive.

In this case, the limit of the general term is 0, so we can conclude that the Divergence Test is inconclusive for the given series.

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