Consider the following series. 1 9 1 18 1 27 1 36 1 45 Determine whether the series is convergent or divergent. Justify your answer. Converges; the series is a constant multiple geometric series. Converges; the limit of the terms, an, is 0 as n goes to infinity. Diverges; the limit of the terms, an, is not 0 as n goes to infinity. Diverges; the series is a constant multiple of the harmonic series. If it is convergent, find the sum. (If the quantity diverges, enter DIVERGES.)

Diverges; the limit of the terms, an, is not 0 as n goes to infinity

Step-by-step explanation:

Convergence of Infinite Series

The series is given as the terms

1 , 9 , 1 , 18 , 27 , 1 , 36 , 1 , 45 ...

For a series to be convergent, the limit of the general term an when n goes to infinity must be 0. In such cases, the sum of all terms tends to a fixed value.

Our series has two clearly different sequences, which define it as a piecewise general term:

If n is odd, then

Otherwise, an is an arithmetic series starting from 9 with a common difference of 9, that is

if n is even.

If we take the limit as n goes to infinity, we don't know which one should be selected, thus we must assume that any of them is a correct option, therefore both of them should converge.

The limit of the function

is 1 when n goes to infinity. Since it's not 0, this piece is not convergent and the entire series isn't either.

But it goes worse because of the limit of the second piece

when n goes to infinity is infinite.

The answer is

Diverges; the limit of the terms, an, is not 0 as n goes to infinity

yes

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