Final answer:
To determine whether the series ∑n=0[infinity]5n4n−2n converges or diverges, we can use the ratio test. The ratio test states that if the absolute value of the ratio of the (n+1)th term to the nth term is less than 1, then the series converges.
Step-by-step explanation:
To determine whether the series ∑n=0[infinity]5n4n−2n converges or diverges, we can use the ratio test. The ratio test states that if the absolute value of the ratio of the (n+1)th term to the nth term is less than 1, then the series converges. Let's apply this test to our series.
First, let's find the nth term of the series. The nth term is given by 5^(n) * 4^(n-2n).
Next, let's find the (n+1)th term by substituting n+1 into the formula. The (n+1)th term is given by 5^(n+1) * 4^(n+1-2(n+1)).
Now, let's calculate the ratio by dividing the absolute value of the (n+1)th term by the absolute value of the nth term: (5^(n+1) * 4^(n+1-2(n+1))) / (5^(n) * 4^(n-2n)).
Simplifying the ratio gives us (5^n * 5 * 4^n * 4^(1-2n)) / (5^n * 4^n) = 20 / 5^n.
Since 20 / 5^n is always less than 1, we can conclude that the series ∑n=0[infinity]5n4n−2n converges.