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Explain Bot To determine whether the series ∑∞,n=2, 1/(n√(n^2−1)) converges or diverges, we can use the Comparison Test. The Comparison Test is a method that compares a given series with another series whose convergence or divergence is already known.
Let's choose the harmonic series ∑∞,n=1, 1/n as our comparison series. The harmonic series is a well-known series that diverges.
To apply the Comparison Test, we need to compare our given series 1/(n√(n^2−1)) with the harmonic series 1/n.
First, we check that all terms of our given series are positive for n ≥ 2. This is true because both n and √(n^2−1) are positive for n ≥ 2.
Next, we take the limit as n approaches infinity of the ratio of the nth term of our given series to the nth term of the harmonic series:
lim(n→∞) [1/(n√(n^2−1))] / [1/n]
To simplify, we can multiply the numerator and denominator of the fraction by n:
lim(n→∞) [n / (n√(n^2−1))] / [1]
Simplifying further, we get:
lim(n→∞) [1 / √(n^2−1)]
Now, let's analyze this limit. As n approaches infinity, the term √(n^2−1) also approaches infinity. Therefore, the limit becomes:
lim(n→∞) [1 / ∞] = 0
Since the limit is zero, and the harmonic series diverges, we conclude that our given series ∑∞,n=2, 1/(n√(n^2−1)) also diverges.
In conclusion, using the Comparison Test, we have determined that the series ∑∞,n=2, 1/(n√(n^2−1)) diverges.