Let x (t) be a periodic function with period T= 10. The fourier series coefficient for this series are denoted by $a_{k}$ , that is
$x (t) = \sum_{k = - \infty}^{\infty} a_{k} e^{j k} \frac{2 π}{T} t$
The same function x(t) can also be considerd as a periodic function with period T'=40. Let $b_{k}$ be the Fourier series coefficient when period is taken as T'.
if $\sum_{k = - \infty}^{\infty} ∣ a_{k} ∣= 16, t h e n \sum_{k = - \infty}^{\infty} ∣ b_{k} ∣$ is equal to

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Solution

Change in only time period or frequency does not change in the value of Fourier serier coefficients.
So, $b_{k} = a_{k}$
$\sum_{k = - \infty}^{\infty} ∣ b_{k} ∣= \sum_{k = - \infty}^{\infty} ∣ a_{k} ∣= 16$

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Change in only time period or frequency does not change in the value of Fourier serier coefficients. So, bk=ak ∑∞k=−∞∣bk∣=∑∞k=−∞∣ak∣=16

Change in only time period or frequency does not change in the value of Fourier serier coefficients.
So, $b_{k} = a_{k}$
$\sum_{k = - \infty}^{\infty} ∣ b_{k} ∣= \sum_{k = - \infty}^{\infty} ∣ a_{k} ∣= 16$