Question

Limit Comparison TestIn Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))

Accepted Answer

Identify the given series: $$ \sum_{n=1}^{\infty} \frac{n(n + 1)}{(n^2 + 1)(n - 1)} . We $$want to determine if it converges or diverges using the Limit Comparison Test. Choose a comparison series $$ b_n $$ that resembles the behavior of $$ a_n = \frac{n(n + 1)}{(n^2 + 1)(n - 1)} $$ for large $$ n . $$Simplify the dominant terms: numerator behaves like $$ n^2 $$, denominator behaves like $$ n^3 $$, so $$ a_n $$ behaves like $$ \frac{n^2}{n^3} = \frac{1}{n} . $$Thus, choose $$ b_n = \frac{1}{n} . $$Compute the limit $$ L = \lim_{n 	o \infty} \frac{a_n}{b_n} = \lim_{n 	o \infty} \frac{\frac{n(n + 1)}{(n^2 + 1)(n - 1)}}{\frac{1}{n}} = \lim_{n 	o \infty} \frac{n(n + 1) \cdot n}{(n^2 + 1)(n - 1)} . $$Simplify the expression inside the limit and evaluate $$ L . If$$ $$ L  is a $$finite positive number (i.e., $$ 0 \u003c L \u003c \infty $$), then $$ a_n $$ and $$ b_n $$ have the same behavior regarding convergence. Since $$ \sum \frac{1}{n}  ($$the harmonic series) diverges, use the result of the Limit Comparison Test to conclude whether the original series converges or diverges.

Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))

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Key Concepts

Limit Comparison Test

Behavior of Rational Functions for Large n

Convergence of p-Series