Find <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAtom-ORD">

Cierra Castillo 2022-07-08 Answered
Find
lim n n 2 k = 1 n 1 ( n 2 + k 2 ) ( n 2 + k 2 + n ) .
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Answers (1)

Jaruckigh
Answered 2022-07-09 Author has 11 answers
For the last integral consider the indefinite integral
I = 1 x 2 + 1 + 1 d x = x 2 + 1 1 x 2 d x
Here, you can use the hyperbolic trig substitution x = sinh t for the first summand in the numerator which yields
x 2 + 1 x 2 d x = coth 2 t   d t = t coth t + C
which after substituting back yields an antiderivative for I
I = sinh 1 x x 2 + 1 1 x + C
Note that this function is, as expected, regular at x = 0, unlike the one in the 2nd equation above.
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