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\def\d{\displaystyle}
\input epsf
\def\d{\displaystyle}
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\centerline {\largerbold Problems, Week 7}

\bigskip

%10.3 integral test, estimates
%10.4 comparison tests

1.~Determine which of the following statements are $\underline{\rm
always}$ true. If a statement is false, give an example to show that
it fails. 
\smallskip
a) If a series doesn't converge, then its $n^{\rm th}$ term doesn't
approach 0 as $n$ approaches $\infty$.
\medskip
b) If the $n^{\rm th}$ term of a series does not approach 0 as n
approaches $\infty$, then the series diverges.
\medskip
c) If all of the partial sums of a series are less than some constant
$L$ , then the series converges.
\medskip
d) If a series converges, then there is a constant $L$ such that all
of its partial sums are less than $L$ .

\bigskip

\settabs 2\columns
2.~Determine which of the following series converge and which diverge.
\medskip
\+a) $\d{ \sum_{n=1}^{\infty} {{2^n + 1} \over {3^n -1}}}$
&b) $\d{ \sum_{n=1}^{\infty} {{\sqrt{2^n+1}} \over  
{\sqrt{3^n-1}}}}$\cr
\+c) $\d{ \sum_{n=1}^{\infty} {{3^{-n}+4^n} \over
{3^n+2^{-n}}}}$
&d) $\d{ \sum_{n=1}^{\infty} {{2^n+n^4} \over
{3^n+n^2}}}$\cr
\bigskip

3.~a) Suppose that $a_n \ge 0$ for all $n$, and that
$\d{\sum_{n=1}^{\infty} a_n}$ converges. Show that
$\d{\sum_{n=1}^{\infty}  a_n^2 }$ must also converge.

Hint: look at the $a_n$'s: they must get small. {\sl Compare} the
size of a small [positive] number and its square.)

\medskip

b) Give an example to show that $\d{\sum_{n=1}^{\infty} a_n^2 }$ can
converge even though $\d{\sum_{n=1}^{\infty} a_n }$ does not
converge.

\bigskip

4.~Consider the sum $$\sum_{k=1}^\infty {1\over{k^3 +\sqrt{k}}} .$$

\smallskip

a) Explain why this sum converges.

\medskip

b) Find $K$ so that $\d{\sum_{k=K}^\infty {1\over{k^3 +\sqrt{k}}}< .0001}$.

\medskip

c)  Find $\d{\sum_{k=1}^\infty {1\over{k^3 +\sqrt{k}}}}$ to within an
accuracy of $\pm .0001$. 

\bigskip

5.~Suppose $\d{z_1 = {1\over 2}}$ and that $\d{z_{n+1} = \left( z_n
\right)^2}$ when $n \ge 1$. Prove either that $\d{\sum_1^\infty z_n}$
diverges, or (if it converges) find its sum to 10 decimal place
accuracy.               

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