Question

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.)

Answer 1

Answer:

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

$a_n=1$

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

$a_n=9+9(n/2-1)=4.5n$

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

$a_n=1$

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

$a_n=4.5n$

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