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Use the ratio test to determine whether the series is convergent or divergent.

1+3/1x2x3+5/1x2x3x4x5+....
a.
convergent
b.
divergent

1 Answer

1 vote

Final answer:

Using the ratio test, the series 1 + 3/1x2x3 + 5/1x2x3x4x5 + ... is found to be convergent because the limit of the ratio of successive terms is 0, which is less than 1. The correct answer is a. convergent.

Step-by-step explanation:

To determine whether the series is convergent or divergent using the ratio test, we'll examine the limit of the absolute value of the ratio of subsequent terms in the series.

If this limit is less than 1, the series converges; if it is greater than 1, or if the limit does not exist, the series diverges.

Consider the general term of the series a_n, where a_n = (2n-1) / (1 x 2 x 3 x ... x (2n-1)).

The next term is a_(n+1) = (2(n+1)-1) / 1 x 2 x 3 x ... x (2n-1) x 2n x (2n+1), which simplifies to (2n+1) / (1 x 2 x 3 x ... x (2n+1)).

To apply the ratio test, we look at the limit as n approaches infinity of |a_(n+1)/a_n|.

After simplifying the ratio a_(n+1)/a_n, we notice that almost all terms cancel out, leaving us with the limit as n approaches infinity of |(2n+1)/(2n x (2n+1))|, which simplifies to 1/(2n).

The limit of 1/(2n) as n approaches infinity is 0. Since 0 is less than 1, by the ratio test, the series is convergent.

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