The divergence test applied to the series ∑n=1 to ∞ 3n/(8n+9) tells us that the series converges or diverges?
I got that it was divergent because it was undefined at infinity, is my answer right?
well, 3n/(8n+9) -> 3/8 as n->∞, so if that's what you meant to say, then you're off.
But the series clearly diverges, since the terms do not tend to zero.
To determine if the series ∑n=1 to ∞ 3n/(8n+9) converges or diverges, let's apply the divergence test.
The divergence test states that if the limit of the nth term of a series is not equal to zero, then the series diverges.
In this case, let's find the limit of the nth term as n approaches infinity:
lim (n→∞) (3n/(8n+9))
To simplify the expression, divide both the numerator and denominator by n:
lim (n→∞) (3/(8+(9/n)))
As n approaches infinity, the term 9/n approaches zero. Therefore, the limit simplifies to:
lim (n→∞) (3/8) = 3/8
Since the limit is not equal to zero, the divergence test tells us that the series diverges.
So, your answer that the series is divergent is correct.
To determine the convergence or divergence of a series, you can apply the divergence test. According to the divergence test, if the limit of the terms of a series does not equal zero, then the series must diverge.
In the given series ∑n=1 to ∞ 3n/(8n+9), let's examine the limit of the terms:
lim (n → ∞) (3n/(8n+9))
To evaluate this limit, we can divide every term in the numerator and denominator by n:
lim (n → ∞) (3n/n)/(8n/n + 9/n)
= lim (n → ∞) (3/8 + 9/n)
As n approaches infinity, the term 9/n approaches zero. Therefore, the limit simplifies to:
lim (n → ∞) (3/8)
Since the limit is a constant value (3/8), it is not equal to zero. Hence, according to the divergence test, the series ∑n=1 to ∞ 3n/(8n+9) diverges.
Therefore, your answer is incorrect, as the series is divergent.