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\centerline{\bf Math 366, Midterm Exam}
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This is a 100 point exam. 
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1. (20 points) True of False. (No proof or counter example needed). 

(a) The limit function of a sequence of continuous functions is always 
continuous.   
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(b) If $\{f_n(x)\}$ and $\{g_n(x)\}$ converge uniformly on a set $E$, 
then their sum $\{f_n(x)+g(x)\}$ also converges uniformly on $E$. 
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(c) A differential function is always equal to its Taylor expansion 
in the interval of convergence. 
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(d) A continuous function in an interval is differentiable at least 
at one point. 
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2. (20 points) 
(a) Define PRECISELY the equicontinuity for a sequence of functions in $E$. 
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(b) Define PRECISELY the Fourier series of a $2\pi$ periodic function. 
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(c) State PRECISELY the Weierstrass Approximation Theorem. 
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(d) State PRECISELY the Bessel's inequality. 
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3. (10 points) Let 
$$f_n(x)=\frac{x^2}{1+nx}.$$ 
Is $\{f_n(x)\}$ convergent uniformly in [0, 1]? 
Verify your answer. 
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4. (10 points) Suppose that $\{f_n(x)\}$ is a sequence of 
uniformly bounded Riemann integrable functions on $[0, 1]$. Let 
$$F_n(x)=\int_0^xf_n(t)dt\quad 0\le x\le 1.$$ 
Prove that there exists a subsequence $\{F_{n_k}\}$ which converges 
uniformly in $[0, 1]$. 
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5. (10 points) Calculate the following limit: 
$$\lim_{x\to 0}(\frac{1+e^x}{2})^{\frac 1x}.$$ 
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6. (15 points) Calculate the Fourier series of the function 
$f(x)=x^2$ in $(-\pi, \pi)$ and then find the sum 
$$\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2}.$$  
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7. (15 points) (a) Suppose $f$ is continuous on $[0, 1]$ and
$$\int_0^1f(x)x^ndx=0, \quad n=0, 1, 2, \cdots.$$
Prove that $f(x)\equiv0$ in $[0, 1]$.
 
(b) Suppose $g$ is continuous on $[0, 1]$ and
$$\int_0^1g(x)x^ndx=0,\quad n=1998, 1999, 2000,  \cdots.$$
Prove that $g(x)\equiv0$ in $[0, 1]$.
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