Example of Comparison Tests for Positive Term Series ∑ a_n

Example: Determine convergence or divergence of the following series.

∞
∑ (√n + 1) / (4n + 3)
n =1

Strategy: Try the limit comparison test.

Let a_n = (√n + 1) / (4n + 3) , b_n = 1 /√n

lim a_n / b_n = √n (√n + 1) / (4n + 3) = ( 1 + 1/n) / (4 + 3/n) = 1/4

n → ∞

Now b_n is a p-series (1/n)^p with p = ½; so b_n diverges

Hence by the limit comparison test a_n diverges. (result)

Example: Determine convergence or divergence of the following series.

∞
∑ sin² n / (n² + 1)
n =1

Strategy: Try a comparison test.

Let a_n = sin² n / (n² + 1), b_n = 1/( n² + 1) c_n = 1/ n²

Now c_n converges since it is the p-series with p = 2

And since b_n ≤ c_n for all n , b_n also converges.

Now sin² n / (n² + 1 ≤ 1/ (n² + 1) for all n

∞

So a_n ≤ b_n for all n. Therefore the series ∑ sin² n / (n² + 1) converges. (result)
n =1

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