LIM (1 / (n ^ 2 + 4) ^ 1 / 2 +... + 1 / (n ^ 2 + 4N ^ 2) ^ 1 / 2) n - > infinity | Math enthusiast | Mathematical art

LIM (1 / (n ^ 2 + 4) ^ 1 / 2 +... + 1 / (n ^ 2 + 4N ^ 2) ^ 1 / 2) n - > infinity

1/(n^2+4)^1/2+...+1/(n^2+4n^2)^1/2)
=∑1/n{1/√[1+4(k/n)^2]}
So, according to the definition of derivative
Original limit = Lim ∑ 1 / N {1 / √ [1 + 4 (K / N) ^ 2]} = ∫ 1 / √ [1 + 4x ^ 2] DX integral range 0 to 1
Let x = 1 / 2tana, a belong to [0, arctan2]
∫ (0 to 1) 1 / √ [1 + 4x ^ 2] DX
=1 / 2 ∫ (0 to arctan2) secada
=1 / 2 ln | SecA + Tana | 0 to arctan2
=(1/2)ln(2+√5)

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