%Summation of Series
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\Large
Summation of Series
\footnote{Copyright \copyright 2002 Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}
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\begin{enumerate}
\item Enter the commands: \\
\verb& syms x k& \\
\verb& format long&
\item Enter: \verb& symsum(.5^k,0,inf)&\\
What kind of series is this? Is the result of the
computation an approximation or is it exact? How fast did it produce an
value for this infinite series? Do you think it was done
numerically or symbolically?
\item Enter: \verb& symsum(.5^k, 0, 10) & followed by \verb& double(ans)&.\\
Here \verb$n = 10$. Increase $n$ gradually until 5 decimal
places of accuracy are reached.
\item Enter: \verb& symsum(.99^k, 0, inf) & and \verb& symsum(.99^k, 0, 10) & \\
Again increase \textsf{n} until 5 decimal places of accuracy
are reached. Compare this with the value of $n$ in the
previous computation, i.e.,\ what is the difference and what
causes it?
\item Try to repeat the process used in \#2 and \#3 for the
series ${\displaystyle \sum_{k = 1}^\infty
\frac{1}{k^{1.1}} }$.\\
(Type: \verb& symsum(k^-1.1,1,inf) & )\\
How fast was this computation? Is the answer exact or approximate?
Was it done symbolically or numerically?
Using an integral estimate (by hand), how many terms are
needed for 5 decimal places accuracy? Try to sum this many terms
and obtain a decimal approximation.
\item Try to guess what the results of the command:
\verb& symsum(x^k/sym('k!'), k, 0, inf) &
\\will be, then enter it. Think about how amazing this
computation is.
\item Prepare a brief (\verb$< $1 page) written report describing
what happened and answering the questions.
Use complete sentences and standard mathematical notation.
Do {\bf not} get a printout.
\end{enumerate}
\vfill
\noindent
\textsf{Symbolic summation and rate of convergence are considered.
For series to be useful for numerical calculations,
convergence must be relatively fast. Series which converge
slowly are only useful in symbolic computations. Computer algebra
systems, such as in \textsc{Matlab}, can perform symbolic computations.
}
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