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Exercises 8.5 Exercises
1.
2.
Use the Weierstrass M-test to prove that the exponential series
\begin{equation*}
\sum_{n=0}^\infty \frac{x^n}{n!}
\end{equation*}
converges uniformly on every interval \([-R,R]\text{.}\)
3.
4.
For each \(n \in \N\text{,}\) define
\begin{equation*}
f_n(x)=
\begin{cases}
4n^2x & \text{if } 0 \le x \le \frac{1}{2n},\\
4n-4n^2x & \text{if } \frac{1}{2n} \lt x \le \frac{1}{n},\\
0 & \text{if } \frac{1}{n} \lt x \le 1.
\end{cases}
\end{equation*}
Show that \(f_n \to 0\) pointwise on \([0,1]\text{,}\) but \(\int_0^1 f_n = 1\) for every \(n\text{.}\)
5.
Define
\begin{equation*}
F(x)=\int_1^x \frac{dt}{t}
\qquad (x \gt 0).
\end{equation*}
Use the derivative formula for \(\ln x\) to prove that \(F(x)=\ln x\) for all \(x \gt 0\text{.}\)
6.
