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\def\d{\displaystyle}
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\centerline {\largerbold Problems, Week 9}

\bigskip

%10.8, 10.9, 10.10 Power series. Function representation. Taylor
%series. Next should be a sample exam. 

1.~Suppose $f(x) = \exp (x^3)$. Find $T_4 (x)$ when $a=0$. (There's
got to be a better way!)

\bigskip

2.~Find the radius of convergence and the interval of convergence for
each of the following power series. 

\settabs 4 \columns
\+$\d{\qquad\sum_{n=1}^{\infty} {{(-1)^n n x^{2n}}}}$
&$\d{\qquad\sum_{n=1}^{\infty} {{x^{2n+1}} \over {(2n+1)!}}}$
&$\d{\qquad\sum_{n=1}^{\infty} {{x^n} \over {(n!)^2}}}$
&$\d{\qquad\sum_{n=1}^{\infty} {{n! x^n}}}$\cr
\bigskip

3.~Suppose $y=f(x)$ is a solution of the differential equation
$$ y' = y + x$$ which satisfies the initial condition $y(0)= 1$ . 

\smallskip

a) What is $f'(0)$? (Use the differential equation.) What is $f''(0)$?
(Differentiate the differential equation and use the information you
already have.) What is $f'''(0)$? (Do it again!) Uhhhh .... maybe you
could now even tell me what $f^{(46)}(0)$ is.  

\medskip

b) What is the Taylor series for $f$ centered at $a=0$ ? (Be a bit
careful. The first few terms are anomalous but the others follow a
fairly simple pattern.)

\medskip

c) If you assume that the Taylor series for $f$ centered at $a=0$
converges to $f$ and that Taylor series are unique, can you identify
$f$ using examples you know?  Try to verify directly that the function
you identified as $f$ is actually a solution of the initial value
problem given in part a).

\bigskip

4.~Find $T_3(x)$ for the function $f(x)= \sqrt{16+x}$ and $a=0$.  Use
$T_3(x)$ to $\underline{ \rm compute}$ an approximate value of
$\sqrt{17}$. First decide the value of $x$ you should take to compute
$\sqrt{17}$. Then find an explicit formula for the error. Find a very
simple overestimate for the error, and $\underline {\rm compute}$ this
overestimate. Finally, compute $\sqrt{17}$ using some method you think
is reliable and check that the numbers you've already gotten are
consistent with what you have computed.

\bigskip

5.~Suppose we want to replace $\sin x$ by $\d{x - {{x^3}
\over {3!}}}$ in some interval centered around $0$ (that is, an
interval like $-A \le x \le A$ with $A > 0$). Find an $A$ so that the
error committed doing such a replacement will be no more than $\pm
.001$. 

\medskip

Note: There is {\it no} unique correct answer to this question. The
difficulty lies in making a convincing error estimate, an estimate to
show that replacing $\sin x$ by this cubic (third degree) polynomial
in the interval you've chosen satisfies the desired error tolerance.
By the way, it is not accidental that the polynomial just happens to
be $T_3 (x)$ [or is it $T_4(x)$?] for $\sin x$ when $a=0$.

\bigskip

6.~For parts a), b) and c), find the Taylor series for each function
about the indicated point $a$, and compute the interval of
convergence. Then do part d).

\smallskip

a) $\d{f(x)= {1 \over {1-x}}}$ and $a=0$ (There is a direct
computation and a short cut. Just do the short cut this time.)

\medskip

b) $\d{g(x) = {1 \over {1+x^2}}}$ and $a=0$ (Easy way: use
part a).)

\medskip

c) ${h(x) = \arctan x}$ and $a=0$ (Easy way: use b) and
notice that $\d{\arctan x = \int_0^x {1 \over {1+t^2}} \,
dt}$.)

\medskip

d) What is the value of the sixteenth derivative of $\arctan x$ at
$x=0$? What about the value of the seventeenth derivative there? 

\bigskip

7.~Write (with summation notation) a finite sum (involving only
integers and their products and quotients) which will approximate
$e^{-.4}$ with an error less than $\pm
10^{-\left(10^{10}\right)}$.

\medskip

Note: You are $\underline{\rm not}$ asked to actually compute the
approximation, but you are asked to assert that one exists and to
write it. There is {\it no} unique correct answer to this
question. One method of attack might be to write the approximation as
$T_n (-.4)$, where $T_n (x)$ is the $n^{\rm th}$ Taylor polynomial for
$e^x$ centered at $a=0$. The difficulty lies in making a convincing
error estimate, an estimate to show that your sum (with the $n$ you
will specify!) is actually within the desired tolerance of $e^{-.4}$.

\bigskip

8.~Consider the function $$F(x) = {{\sin x} \over {1-x}}$$ $F(x)$ has
a Taylor series (of the form $\d{\sum_{j=0}^{\infty} {{F^{(j)} (0)}
\over {j!}} x^j}$). Compute the terms of this Taylor series up to (and
including) the $x^5$ term.

\medskip

Hint: While a direct computation is possible (by differentiating $F$
repeatedly and then evaluating the derivatives), it is {\it very}
unappealing. You should now know series for both $\sin x$ and
$\d{1 \over {1-x}}$. Just multiply the series together. Try
to do no unnecessary work: you just need to multiply and add terms
whose degree is $\le 5$.  This should not be very hard.

\bigskip

9.~Taylor series centered at $0$ may be found for these functions by a
long method or by a short method. Describe both methods, and then use
the short one.

\smallskip

a) $f(x)= x^3 e^x$

\medskip

b) $f(x) = \ln \, (1+x^2)$

\medskip

c) $\d{f(x) = \cases{{\d{{\sin x}\over x}}, &if $x \ne
0$\cr 0 ,& $ x = 0$\cr}}$

\bigskip

10.~a) Can you find a power series centered at $0$ which converges at
$x=3$ and diverges at $x=-2$? Give an example of one if you can, or
give a reason why if you can't. 

\medskip

b) Can you find a power series centered at $0$ which converges
$\underline{\rm only}$ at $x=0$? Give an example of one if you can, or
give a reason why if you can't. 

\medskip

c) Can you find a power series centered at $0$ which converges
$\underline{\rm only}$ for $x$'s in the interval $-3 \le x < 3$ (read
carefully!)? Give an example of one if you can, or give a reason why
if you can't.

\bigskip

11.~What is 
$$\lim_{x \to 0} {{\arctan {(x^2)} - (\arctan x)^2} \over {x^4}}$$
\rightline{\sevenrm Don't use l'H\^opital's rule four times! Please
don't.}


\bigskip

\settabs 2 \columns

12.~Use series techniques to compute each of these integrals to an
accuracy of $\pm .0001$. Explain {\it why} your accuracy is achieved
by estimating the error.

\smallskip

\+ $\d{\qquad \int_0^1 \sin (x^2)\; dx}$
& $\d{\qquad \int_0^{1 \over 2} {{dx} \over {1-x^3}}}$\cr

\bigskip

13.~In certain areas of theoretical computer science, the idea of a
{\sl GENERATING FUNCTION} is very important. Here's the definition:

\medskip

\centerline{ $F$ is a $\underline{\sl GENERATING\ FUNCTION}$ for a
sequence $\{a_n\}_{n=0}^{\infty}$ if $\d{F(x) = \sum_{n=0}^{\infty}
a_n x^n}$.}

\medskip

For example, the exponential function (just $e^x$) is the generating
function for the sequence $\d{\left\{ {1\over {n!}}
\right\}_{n=0}^{\infty}}$. Recognition of the generating functions of
certain obvious sequences is very important in a number of
applications. Below is a list of sequences and a list of
functions. Can you match the sequences with their generating
functions?

\medskip

\+$\underline{\rm Sequences:}$
&$\underline{\rm \vphantom{q} Functions:}$\cr
\smallskip
\+a) $\{1, 1, 1, \ldots \ $(All 1's)$ \}$
&$\alpha$) $\d{{1+x}\over {(1-x)^3}}$\cr
\smallskip
\+b) $\{0, 1, 2, 3, \ldots , n, \ldots \}$
&$\beta$)  $\d{\ln\, \left(1\over{1-x}\right) }$\cr
\smallskip
\+c) $\{0, 1, 4, 9, \ldots , n^2, \ldots \}$
&$\gamma$)  $\d{1 \over {1-2x}}$\cr
\smallskip
\+d) $\d{\{0, 1, {1\over 2}, {1\over 3}, \ldots , {1 \over n},
\ldots \} }$
&$\delta$) $\d{\int_0^x {{\ln
\left(1\over{1-t}\right)} \over t} \, dt}$\cr
\smallskip
\+e) $\d{\{0, 1, {1\over 4}, {1\over 9}, \ldots , {1 \over {n^2}},
\ldots \} }$
&$\epsilon$)  $\d{1 \over {1-x} }$\cr
\smallskip
\+f) $\{1, 3, 3, 1,\ $(All the rest 0's)$ \}$
&$\mu$)  $\d{x \over {(1-x)^2} }$\cr
\smallskip
\+g) $\{1, 2, 4, 8, \ldots , 2^n, \ldots \}$
&$\nu$)  $\d{(1+x)^3}$\cr
\medskip
Note: Generating functions turn out to be a very convenient and
powerful way to describe and manipulate sequences. You'll see them in
probability, also. 


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