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Let x(t) be a periodic function with period T = 10. The Fourier series coefficients for this series are denoted by $${a_k}$$ , that is
$$x\left( t \right) = \sum\limits_{k = - \infty }^\infty {{a_k}{e^{jk{{2\pi } \over T}t}}} .$$
The same function x(t) can also be considered as a periodic function with period T' = 40. Let bk be the Fourier series coefficients when period is taken as T'. If $$\sum\limits_{k = - \infty }^\infty {\left| {{a_k}} \right|} = 16,$$    then $$\sum\limits_{k = - \infty }^\infty {\left| {{b_k}} \right|} $$  is equal to
For a function g(t), it is given that
$$\int\limits_{ - \infty }^{ + \infty } {g\left( t \right){e^{ - j\omega t}}dt = \omega {e^{ - 2{\omega ^2}}}} $$    for any real value $$\omega $$ . If $$y\left( t \right) = \int\limits_{ - \infty }^t {g\left( \tau \right)} d\tau ,\,{\rm{then}}\,\int\limits_{ - \infty }^{ + \infty } {y\left( t \right)} dt$$       is. . . . . . . .
The value of $$\mathop {\lim }\limits_{{\text{x}} \to \infty } {\left( {1 + {{\text{x}}^2}} \right)^{{{\text{e}}^{ - {\text{x}}}}}}$$   is
$$\mathop {{\text{Lim}}}\limits_{{\text{x}} \to \infty } \left( {\frac{{{\text{x}} + \sin {\text{x}}}}{{\text{x}}}} \right)$$    equal to
If $$F\left( s \right) = L\left| {f\left( t \right)} \right| = {K \over {\left( {s + 1} \right)\left( {{s^2} + 4} \right)}},$$       then $$\mathop {\lim }\limits_{t \to \infty } f\left( t \right)$$  is given by
If {x} is a continuous, real valued random variable defined over the interval (-$$\infty $$, +$$\infty $$) and its occurrence is defined by the density function given as:
$${\text{f}}\left( {\text{x}} \right) = \frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}$$     where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integral $$\int_{ - \infty }^{\text{a}} {\frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}{\text{dx}}} $$
Consider an exothermic reaction, $$A\mathop {\mathop \rightleftharpoons \limits_{{k_{ - 1}}} }\limits^{{k_1}} I$$
As the temperature increases
What is the value of $$\mathop {\lim }\limits_{{\text{x}} \to \frac{\pi }{4}} \frac{{\cos {\text{x}} - \sin {\text{x}}}}{{{\text{x}} - \frac{\pi }{4}}}$$
The value of $$\mathop {\lim }\limits_{{\text{x}} \to 8} \frac{{{{\text{x}}^{\frac{1}{3}}} - 2}}{{\left( {{\text{x}} - 8} \right)}}$$
For a complex number z, $$\mathop {\lim }\limits_{{\text{z}} \to {\text{i}}} \frac{{{{\text{z}}^2} + 1}}{{{{\text{z}}^3} + 2{\text{z}} - {\text{i}}\left( {{{\text{z}}^2} + 2} \right)}}$$     is