5.6 Extending the definition of integrability  (Page 2/2)

1. Use [link] , [link] , [link] , and properties of limits to prove the preceding theorem.
2. Let $f$ be defined and improperly-integrable on $(a,b).$ Show that, given an $ϵ>0,$ there exists a $\delta >0$ such that for any $a<a^{\text{'}}<a+\delta$ and any $b-\delta <b^{\text{'}}<b$ we have $|{\int }_a^{a^{\text{'}}}f|+|{\int }_{b^{\text{'}}}^{b}f|<ϵ.$
3. Let $f$ be improperly-integrable on an open interval $(a,b).$ Show that, given an $ϵ>0,$ there exists a $\delta >0$ such that if $(c,d)$ is any open subinterval of $(a,b)$ for which $d-c<\delta ,$ then $|{\int }_c^{d}f|<ϵ.$ HINT: Let $\left\{x_i\right\}$ be a partition of $[a,b]$ such that $f$ is defined and improperly-integrable on each subinterval $(x_{i-1},x_i).$ For each $i,$ choose a ${\delta }_i$ using part (b). Now $f$ is bounded by $M$ on all the intervals $[x_{i-1}+{\delta }_i,x_i-{\delta }_i],$ so $\delta =ϵ/M$ should work there.
4. Suppose $f$ is a continuous function on a closed bounded interval $[a,b]$ and is continuously differentiable on the open interval $(a,b).$ Prove that $f^{\text{'}}$ is improperly-integrable on $(a,b),$ and evaluate ${\int }_a^b{f}^{\text{'}}.$ HINT: Fix a point $c\in (a,b),$ and use the Fundamental Theorem of Calculus to show that the two limits exist.
5. (Integration by substitution again.) Let $g:[c,d]\to [a,b]$ be continuous on $[c,d]$ and satisfy $g\left(c\right)=a$ and $g\left(d\right)=b.$ Suppose there exists a partition $\{x_0<x_1<...<x_n\}$ of the interval $[c,d]$ such that $g$ is continuously differentiable on each subinterval $(x_{i-1},x_i).$ Prove that $g^{\text{'}}$ is improperly-integrable on the open interval $(c,d).$ Show also that if $f$ is continuous on $[a,b],$ we have that
   $${\int }_a^{b}f\left(t\right)dt={\int }_c^{d}f\left(g\left(s\right)\right)g^{\text{'}}\left(s\right)ds.$$
   HINT: Integrate over the subintervals $(x_{i-1},x_i),$ and use part (d).

1. Define $f$ to be the function on $[1,\infty )$ given by $f\left(x\right)={(-1)}^{n-1}/n$ if $n-1\le x<n.$ Show that $f$ is improperly-integrable on $(1,\infty ),$ but that $\left|f\right|$ is not improperly-integrable on $(1,\infty ).$ (Compare this with part (4) of [link] .) HINT: Verify that ${\int }_1^{N}f$ is a partial sum of a convergent infinite series, and then verify that ${\int }_1^N\left|f\right|$ is a partial sum of a divergent infinite series.
2. Define the function $f$ on $(1,\infty )$ by $f\left(x\right)=1/x.$ For each positive integer $n,$ define the function $f_n$ on $(1,\infty )$ by $f_n\left(x\right)=1/x$ if $1<x<n$ and $f_n\left(x\right)=0$ otherwise. Show that each $f_n$ is improperly-integrable on $(1,\infty ),$ that $f$ is the uniform limit of the sequence $\left\{f_n\right\},$ but that $f$ is not improperly-integrable on $(1,\infty ).$ (Compare this with part (5) of Theorem 5.6.)
3. Suppose $f$ is a nonnegative real-valued function on the half-open interval $(a,\infty )$ that is integrable on every closed bounded subinterval $[a,b^{\text{'}}].$ For each positive integer $n\ge a,$ define $y_n={\int }_a^{n}f\left(x\right)dx.$ Prove that $f$ is improperly-integrable on $[a,\infty )$ if and only if the sequence $\left\{y_n\right\}$ is convergent. In that case, show that ${\int }_a^{\infty }f=lim{y}_n.$

1. Let $s$ be a real number. Use the Integral Test toprove that the infinite series $\sum 1/n^s$ is convergent if and only if $s>1.$
2. Let $s$ be a complex number $s=a+bi.$ Prove that the infinite series $\sum 1/n^s$ is absolutely convergent if and only if $a>1.$

Let $f$ be the function on $[1,\infty )$ defined by $f\left(x\right)=1/x.$

1. Use [link] to prove that the sequence $\{{\sum }_{i=1}^N\frac{1}{i}-lnN\}$ converges to a positive number $\gamma \le 1.$ (This number $\gamma$ is called Euler's constant.) HINT: Show that this sequence is bounded above and nondecreasing.
2. Prove that
   $$\sum _{i=1}^{\infty }\frac{{(-1)}^{i+1}}{i}=ln2.$$
   HINT: Write $S_{2N}$ for the $2N$ th partial sum of the series. Use the fact that
   $$S_{2N}=\sum _{i=1}^{2N}\frac{1}{i}-2\sum _{i=1}^N\frac{1}{2i}.$$
   Now add and subtract $ln\left(2N\right)$ and use part (a).