Problem 1
Let \(\{g_n\}\) be a sequence of real valued functions such that \(g_{n+1}(x)\leq g_n(x)\) for each \(x\) in T and for every \(n=1,2,...\). If \(\{g_n\}\) is uniformly bounded on T and if \(\sum f_n(x)\) converges uniformly on T, then \(\sum f_n(x)g_n(x)\) also converges uniformly on T.
Problem 2
Let \({a_n}\) be a decreasing sequence of positive terms. Prove that
the series \( \sum a_n \sin nx\) converges uniformly on \(\mathbb{R}\)
if, and only if, \(na_n\to 0\) as \(n\to\infty\)
Only if direction is easy so we omit the proof here, we only prove the if direction
Suppose
\(na_n\to 0\) as \(n\to\infty\), let \(B_n(x)=\sum\limits_{k=1}^{n}\sin
kx\)
\(\sum\limits_{k=1}^{n}a_k \sin kx=B_n a_{n+1}-\sum\limits_{k=1}^{n}B_k
(a_{k+1}-a_k)\)
\begin{align*}
\bigg|\sum\limits_{k=n+1}^{m}a_k \sin kx\bigg|&=\bigg| B_n
a_{n+1}-B_m a_{m+1}+\sum\limits_{k=n+1}^{m}B_k (a_{k+1}-a_k)\bigg|\\
&=\bigg| B_n a_{n+1}-B_m a_{m+1}+ B(a_{m+1}-a_{n+1}) \bigg|,\;\;\ \\
&\mbox{where}\;min\{B_{n+1},B_{n+2},...,B_{m}\}\leq B \leq
max\{B_{n+1},B_{n+2},...,B_{m}\}\\
&\leq \bigg|(B_n-B) a_{n+1} \bigg|+ \bigg|(B-B_m) a_{m+1} \bigg|\\
&<2Na_{n+1}+2Ma_{m+1}\\
&<\varepsilon
\end{align*}
problem 1 can be proved in a similar way
Problem 3
Given a power series \(\sum_{n=0}^{\infty} a_n z^n\) whose
coefficients are related by an equation of the form
\[a_n+Aa_{n-1}+Ba_{n-2}=0\] (n=2,3,...)
Show that for any \(x\) for which ther series converges, its sum is \[\frac{a_0+(a_1+Aa_0)x}{1+Ax+Bx^2}\]
Problem 4
Show that the binomial series \((1+x)^{\alpha}=\sum_{n=0}^{\infty}
{\alpha \choose k} x^n\) exhibits the following behavior at the points
\(x=\pm 1\)
a) If \(x=-1\), the series converges for \(\alpha\geq 0\) and diverges for \(\alpha<0\)
b)
If \(x=1\), the series diverges for \(\alpha \leq -1\), converges
conditionally for \(\alpha\) in the inteval \(-1<\alpha<0\), and
converges absolutely for \(\alpha\geq0\)
