Consider series Sigma (n=1 to inf) an, where an = (-1)^n / (n^2 + 2n + 3) Comput

Question

Consider series Sigma (n=1 to inf) an, where

           an = (-1)^n / (n^2 + 2n + 3)

Compute L = lim (n-->inf) (|an+1|) / (an)

a) Enter the numerical value of limit L if it converges. Otherwise, state if it diverges to inf, neg inf, or diverges but not to inf or neg inf

b) Which of these statements is true

- Root test says that series converges absolutely

- Root test says series diverges

- Root test says series converges conditionally

- Root test inconclusive but series converges absolutely by another test

- Root test inconclusive but series diverges by another test

- Root test inconclusive but series converges conditionally by another test

Explanation / Answer

The ratio is simply: a[n+1]/a[n] = ((-1)^(n+1) ( (n^2 + 2n + 3)) / ((-1)^n ((n+1)^2 + 2(n+1) + 3)) and you can easily see that both of the latter two factors become closer and closer to 1 as n -> infinity So the limit of the ratio = (-1)^1 and a.) the series converges to -1 b.) Root test says that series converges absolutely This is the correct answer. Please rate with 5 stars

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