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# Week 12 Problem Set A.
## Reading.
Chapter 5.1, 5.2, 5.3
On areas, sigma notation (summation notation), and the definite integral.
## Problems.
### Sigma notation (summation notation).
It is important to get comfortable with these notations! Pay attention to the indices and be careful how to manipulate them.
1. Which of the following express $1+2+4+8+16+32$ in sigma notation?
1. $\sum_{k=1}^{6} 2^{k-1}$
2. $\sum_{k=0}^{5}2^{k}$
3. $\sum_{k=-1}^{4}2^{k+1}$
2. Which of the following express $1-2+4-8+16-32$ in sigma notation?
1. $\sum_{k=1}^{6}(-2)^{k-1}$
2. $\sum_{k=0}^{5}(-1)^{k}2^{k}$
3. $\sum_{k=-2}^{3}(-1)^{k+1}2^{k+2}$
3. Which formula below is not equivalent to the other two?
1. $\sum_{k=2}^{4} \frac{(-1)^{k-1}}{k-1}$
2. $\sum_{k=0}^{2} \frac{(-1)^{k}}{k+1}$
3. $\sum_{k=-1}^{1} \frac{(-1)^{k}}{k+2}$
4. Express the sums below in sigma notation. You can choose your lower limit of summation, so your answer may "look different" depending on your choice.
1. $1+2+3+4+5+6$
2. $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$
3. $1- \frac{1}{2}+ \frac{1}{3}-\frac{1}{4}+ \frac{1}{5}$
5. Suppose that $\sum_{k=1}^{n}a_{k}=-5$ and $\sum_{k=1}^{n}b_{k}=6$. Find the values of the following:
1. $\sum_{k=1}^{n}3a_{k}$
2. $\sum_{k=1}^{n} \frac{b_{k}}{6}$
3. $\sum_{k=1}^{n} (a_{k}+b_{k})$
4. $\sum_{k=1}^{n}(a_{k}-b_{k})$
5. $\sum_{k=1}^{n}(b_{k}-2a_{k})$
### Faulhaber formulas.
In class we derived some of the Faulhaber formulas, these for sums of the form $\sum_{k=1}^{n}k^{p}$ for a positive integer $p$. In general, $$
\sum_{k=1}^{n}k^{p} = \frac{n^{p+1}}{p+1}+\cdots
$$is a degree $p+1$ polynomial in $n$, with leading coefficient $\frac{1}{p+1}$.
Use the Faulhaber formulas to evaluate the following sums:
1. $\sum_{k=1}^{10}k$
2. $\sum_{k=1}^{10}k^{2}$
3. $\sum_{k=1}^{10}\frac{k^{3}}{7}$
4. $\sum_{k=4}^{17}k^{2}$
5. $\sum_{k=12}^{40} 3k - 2k^{3}$
Now, use the method shown in class, try deriving a closed form formula for the sum $\sum_{k=1}^{n} k^{3}$. Hint, start by considering $\sum_{k=1}^{n}(k+1)^{4}$. It will be a bit tedious but not impossible!
### Riemann sum.
Recall for any bounded function $f(x)$ defined over the interval $[a,b]$, a Riemann sum for $f(x)$ over the interval $[a,b]$ is just a sum of the form $$
S_{P}(f)=\sum_{k=1}^{n}f(c_{k})\Delta x_{k}
$$ where $P$ is a partition of the interval $[a,b]$, with $c_{k}$ some sample point in the $k$-th subintervals $[x_{k-1},x_{k}]$, and $\Delta x_{k} = x_{k}-x_{k-1}$ is the width of the $k$-th subinterval.
1. For the function $f(x)=3-x^{2}$ over the interval $[2,5]$, calculate the Riemann sum of $f$ using the partition $P = \{2 < 2.5 < 3 < 4 < 5\}$ with sample points $c_{1} = 2$, $c_{2}=3$, $c_{3}=3.5$, $c_{4}=4.5$.
2. What is the **norm** of the partition $P=\{5 < 5.2 < 6 < 6.9 < 7\}$ for the interval $[5,7]$?
In class, we discussed special kinds of Riemann sum for a function $f(x)$ over the interval $[a,b]$, namely
(1) the right end point Riemann sum with $n$ pieces, given by $$
R_{n}(f) = \sum_{k=1}^{n}f(a + k \Delta x)\cdot\Delta x
$$
(2) the left end point Riemann sum with $n$ pieces, given by $$
L_{n}(f) = \sum_{k=1}^{n}f(a + (k-1) \Delta x)\cdot\Delta x
$$
(3) the midpoint Riemann sum with $n$ pieces, given by $$
M_{n}(f) = \sum_{k=1}^{n} f( a+ ( k-\frac{1}{2} ) \Delta x ) \cdot \Delta x
$$In each of these special Riemann sum, the subintervals all have equal widths $\displaystyle\Delta x = \frac{b-a}{n}$.
These will be useful to carry out actual computations.
### Limits of Riemann sum.
In each of the following, (1) **find $R_{n}(f)$, the right hand endpoint Riemann sum with $n$ pieces** for the given function $f$ over the given interval $[a,b]$, then (2) take the limit as $n\to \infty$ to calculate the area under the curve over $[a,b]$.
1. $f(x) = 1-x^{2}$ over the interval $[0,1]$
2. $f(x) = x^{2}+1$ over the interval $[0,3]$
3. $f(x) = x + x^{2}$ over the interval $[0,1]$
You may find the Faulhaber formulas helpful.
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