1. Find a formula for the general term of the sequence, assuming the pattern continues and that the first term corresponds to , 2. Determine whether the sequence converges or diverges. If it converges, find its limit. 3. Determine whether each series is convergent or divergent. If it is convergent, find its sum.

  1. an = (-1)^(n-1)·(n+2)!/3^n

  2. the sequence diverges

Step-by-step explanation:

Perhaps you're concerned with the sequence ...

  {2, -24/9, 120/27, -720/81, ...}

1. This is neither arithmetic nor geometric. Ratios of terms are -4/3, -5/3, -6/3.

The alternating signs mean one factor of the general term is (-1)^(n-1). The divisors of 3 in the term ratios indicate 3^-n is another factor. The increasing multipliers suggest that a factorial is involved.

If we rewrite the sequence factoring out (-1)^(n-1)/3^n, we have ...

  {6, 24, 120, 720, ...}

corresponding to 3!, 4!, 5!, 6!. This lets us conclude the remaining factor is (n+1)!.

The general term is ...

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2. The magnitude of the factorial quickly outstrips the magnitude of the exponential denominator, so the terms keep getting larger and larger.

The sequence diverges.

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3. No series are provided.

b) well because you get well