Question

Can determine, with sufficient justification, whether an infinite sequence diverges or converges. When an infinite sequence converges, I can determine that limit. I can apply valid reasoning to determine whether a sequence converges when given knowledge about a similar sequence.

Certainly, I'll avoid using mathematical symbols. Here are your questions with caret (^) notation:

Compute the first six terms of the sequence a_n = n^(1-n). If the sequence converges, find its limit.

Determine if the sequence a_n = (n! + 2^n)/(4^n + 5n!) converges or diverges. If the sequence converges, find its limit. State your reasoning.

Answer 1

The first six terms of the sequence a_n = n^(1-n) suggest a trend towards convergence to 0. For the sequence a_n = (n! + 2^n)/(4^n + 5n!), the limit is 1, as n! dominates both the numerator and denominator for large n.

Step-by-step explanation:

Convergence of Infinite Sequences

First, we will compute the first six terms of the sequence a_n = n^(1-n). Remember that any number raised to the power of zero is 1. So, a_1 = 1^(1-1) = 1, a_2 = 2^(1-2) = 1/2, and this pattern continues negatively exponentially:

• a_3 = 3^(-2)
• a_4 = 4^(-3)
• a_5 = 5^(-4)
• a_6 = 6^(-5)

This sequence converges to 0, as larger values of n will make the fraction even smaller.

Now, for a_n = (n! + 2^n)/(4^n + 5n!), notice that for large values of n, the n! terms dominate both the numerator and denominator. This suggests that the limit as n approaches infinity for this sequence is 1, since the ratio of n! to itself is 1, and the remaining terms become negligible.