Question

Find the limit of the sequence of partial sums whose general term is D%7Bn%21%7D" id="TexFormula1" title="a_n=\frac{100^n}{n!}" alt="a_n=\frac{100^n}{n!}" align="absmiddle" class="latex-formula">
10

1

DNE

0

I am torn between the limit not existing and it being = 0.

Thank you so much.

Answer 1

Answer:

0

Step-by-step explanation:

If ∑aₙ converges, then lim(n→∞)aₙ = 0.

Using ratio test, we can determine if the series converges:

If lim(n→∞) |aₙ₊₁ / aₙ| < 1, then ∑aₙ converges.

If lim(n→∞) |aₙ₊₁ / aₙ| > 1, then ∑aₙ diverges.

lim(n→∞) |(100ⁿ⁺¹ / (n+1)!) / (100ⁿ / n!)|

lim(n→∞) |(100ⁿ⁺¹ / (n+1)!) × (n! / 100ⁿ)|

lim(n→∞) |(100 / (n+1)|

0 < 1

The series converges.  Therefore, lim(n→∞)aₙ = 0.