Get the solution to "Evaluating a limit, limt→0+∑n=1∞t1+tn2"

Dominique Pace

Answered question

2022-04-03

Evaluating a limit,

lim_{t \to 0^{+}} \sum_{n = 1}^{\infty} \frac{\sqrt{t}}{1 + t n^{2}}

Jazlyn Mitchell

Beginner2022-04-04Added 14 answers

Note the hyperbolic cotangent identity

\sum_{n = 1}^{\infty} \frac{1}{z^{2} + n^{2}} = \frac{π z \coth (π z) - 1}{2 z^{2}}

Replace t with

t^{2}

for convenience. Then

\sum_{n = 1}^{\infty} \frac{t}{1 + t^{2} n^{2}} = \frac{1}{t} \sum_{n = 1}^{\infty} \frac{1}{t^{- 2} + n^{2}} = \frac{1}{2} [π \coth (\frac{π}{t}) - t]

Observe that

\coth (s) \to 1

s \to \infty

and

t \to 0

, so the limit is

\frac{π}{2}

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