lets-f-0-T-R-such-that-0-T-f-t-2-dt-lt-T-2pi-if-a-n-2-T-0-T-f-t-cos-nt-dt-and-b-n-2-T-0-T-f-t-sin-nt-dt-does-lim-n-0-a-n-a-0-lim-n-0-b-n-0-

Question Number 4714 by 123456 last updated on 28/Feb/16

lets f : [0, T] \to R such that

\underset{0}{\int^{T}} {[f (t)]}^{2} d t < + \infty

ω T = 2 π

if a (n) = \frac{2}{T} \underset{0}{\int^{T}} f (t) \cos (ω n t) d t

and b (n) = \frac{2}{T} \underset{0}{\int^{T}} f (t) \sin (ω n t) d t

does ?

\lim_{n \to 0} a (n) = a (0)

\lim_{n \to 0} b (n) = 0

Commented by prakash jain last updated on 29/Feb/16

I think in this case we can directly substitute

the value of n before evaluating the integral .

So \lim_{n \to 0} a (n) = a (0)

\lim_{n \to 0} b (n) = 0

The given condition \underset{0}{\int^{T}} {[f (t)]}^{2} d t < + \infty implies that limit

exists .