2.Suppose that $Q$ is a refinement of the partition $P$ of the interval $[a, b]$ and that $f$ is a bounded function on $[a, b]$. Show that

$$L(f, P) ≤ L(f, Q) ≤ U(f, Q) ≤ U(f, P)$$.

Derive, as a corollary that $L(f, P_1) ≤ U(f, P_2)$ for all partitions $P_1$ and $P_2$ of $[a, b]$.

3.Define $f$ to be the step function

\[ \left\{

\begin{array}{ll}

1 & -3 \leq x < 0 \\

-2 & 0\leq x \leq 1\\

\end{array}

\right. \]

Show that $f$ is integrable over $[−3, 1]$ and that $\int_{-3}^{1}f=1$.

4.Suppose that $f_n : [a, b] → R$ is integrable on $[a, b]$ for each $n$ and that $f_n → f$ uniformly on $[a, b]$.

Prove that $f$ is integrable on $[a, b]$.

5.Manipulate the geometric series to show that

$\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}= \frac{\pi}{4}$

Note: You’ll need to substitute $x → −x^2$, integrate the result, and plug in the appropriate number. That number is on the boundary of the domain of convergence, though, so you’ll need to use Abel’s theorem to justify the equality.