Name Elliot Gootman
Goals/Objectives To see how to use spreadsheets to find out what happens with infinite sums.
Conjectures/hypothesis/prediction Some series sum up to
a particular number, while others just keep on getting
bigger and bigger without end.
Description of Procedures/methods We found the pattern for each
term of the series, entered the formula for the
pattern into a spreadsheet, and then computed the partial sums (that is, the sums from the beginning to a certain point).
Conclusions(Analysis using technology and then understand WHY it
works). For the series whose nth term is
n/2^n, it seemed that the series summed up to 2. It was hard to tell from the spreadsheet that it would sum up to a
given number, or what that number would be. The related series whose nth term is simply 1/2^n is easier to understand. It sums up to 1. There seemed to be an algebra trick relating the original series to the simpler one, that we could use to prove that the original series summed up to 2.
Comments/Reactions/What did I learn?- Was technology helpful? Since we originally new nothing about series, technology was necessary to sum up the series. To analyze why a series sums up to what it does, it seems to be helpful to know something about a few basic series, and then to compare a given series with the basic series. It seems that some sort of algebra software, or symbolic manipulation software, might be helpful for this.
Alternate methods (what would I have done differently) I could
have started out playing around with algebra, and trying to relate the
given series to a simpler one that I know about. This would have been trial
and error, but that's OK.
Actually, because the demo was running longer than I expected, I forgot to do the most important exploration -
comparing the size of a term with the size of the preceding term. This would have been a great intro to simpler series of the form 1/2^n, etc.
Or, if I knew nothing at all about series, I might just have looked at a simpler series. Suppose we keep the pattern for the denominators, but make all the numerators equal to the first numerator of one. Suppose we keep the pattern for the numerators, but make all the denominators equal to one.
Teaching strategies - (looking at the investigation from both a learning perspective and a teaching perspective). I would have students explore some simpler series first- 1/2^n, 1/3^n.
Extensions (what other questions can I ask or explore?)
click here to see the excel spreadsheet