open a text line to annotate your work is Alt-7. From Eq. 2.5 we use the "sum" function in Mathematica to consider the (finite) sum

Transcription

1 Lecture Our goal here is to begin to learn to use Mathematica while at the same time use it to eplore the concepts in the tet. You are encouraged to confirm the Mathematica results and the mechanical issue of running the program, entering the commands, etc., i.e., run your own session of Mathematica and enter these commands. Mathematica has pretty good built-in Help and you are strngly encouraged to use it to look-up definitions of functions, find new functions, etc. The key stroke instructing Mathematica to perform a command (or evaluate an epression) is the combination shift-enter. The command to open a tet line to annotate your work is Alt-7. From Eq.. we use the "sum" function in Mathematica to consider the (finite) sum In[]:= Out[]= Sum@ ^ n, 8n, 4<D Alternatively we can enter the same epression with In[]:= Out[]= Sum@ ^ n, 8n,, 4<D ote the general feature that Mathematica always begins the names of pre-installed functions with capital letters, puts the arguments in square brackets, []'s, and uses curly brackets, {}'s, to define parameters for the function; here to sum over the inde n from to 4 (i.e., {n,4} = {n,,4}). ow define the infinite version (note how Mathematica likes to label the "quantity" infinity; it can also be entered with the symbol ). In defining a function we use the underscore, _, to label the arguments (on the left-hand-side) and the symbol ":=" (and not just "=") In[3]:= S@_D := Sum@ ^ n, 8n, Infinity<D To ask Mathematica to perform the sum we simply type the name of the function (with no underscoring), In[4]:= S@D Out[4]= - + This is a common result that Mathematica does not simplify automatically to the epected analytic epression. If you epect that simplification is possible, you can ask Mathematica to look for it. There is a handy shorthand if you want to perform an operation of the previous epression - just represent it by the % symbol. In[]:= Simplify@%D Out[]= - + Given the current order in which the commands have been evaluated, this last command is the same as In[6]:= Simplify@S@DD Out[6]= - + This the result of Eq.(.). For comparison we can also evaluate the sum starting with, the power zero, as in Eq.(.9), the so-called geometric series

2 Lec_7_8.nb This the result of Eq.(.). For comparison we can also evaluate the sum starting with, the power zero, as in Eq.(.9), the so-called geometric series In[7]:= Out[7]= In[8]:= ^ n, 8n,, 4<D Sum@ ^ n, 8n,, <D Out[8]= - An essential strength of the Mathematica software is the ability to analytically perform this infinite summation for a general variable, as here. On the other hand you have to be careful not to be misled by the result. In particular, for < the infinite series defines the function indicated. However, while the functions is well defined for >, the infinite series itself is divergent. We can study the issue of convergence by looking at the behavior of the finite (truncated) series and then the remainder. First define the finite series as a function of both and. In[9]:= In[]:= _D := Sum@ ^ n, 8n,, D So let's use our plotting ability to check the behavior of this function as a function of for fied values of, e.g., =.. In[]:= Plot@@,.D, 8,, <D..9 Out[]=.9 Let's clean this up a bit and provide labels.

3 Lec_7_8.nb In[]:= 8,, <, PlotRange 8, <, AesLabel 8, <, AesOrigin -> 8, <D. Out[]=. It clearly converges by around. ow lets try a few values of. @,.9D<, 8,, <, PlotRange 8, <, AesLabel 8, <, AesOrigin -> 8, <D. Out[3]=. Clearly both the number the series converges to varies with and the actual converge properties vary with the value, but the details are difficult to see in this plot. To clean up the plot we etend the limits and the aes. 3

4 4 @,.9D<, 8,, <, PlotRange 8, <, AesLabel 8, <, AesOrigin -> 8, <D 8 6 Out[4]= However, note that In[]:= Plot@@,.D, 8,, <, PlotRange 8, <, AesLabel 8, <, AesOrigin -> 8, <D 4 3 Out[]= So for > we see the finite sum diverging as gets large. To focus on the question of convergence/divergence can look at the remainder of all the terms in the infinite series after the th term, R = S. In[6]:= In[7]:= R@_, _D := Sum@ ^ n, 8n, +, Infinity<D R@, D + Out[7]= - D + R@, D + + Out[8]= - H- + L - + A bit messy so simplify.

5 Lec_7_8.nb In[9]:= D + R@, DD Out[9]= - + The epected analytic result for the full sum. Consider the remainder numerically. In[]:= Out[]= R@,.3D Mathematica has trouble recognizing the convergence but we can look for it graphically In[]:= Plot@R@,.3D, 8,, <, PlotRange 8, <, AesLabel 8, R<, AesOrigin -> 8, <D Unset::wrsym : Symbol is Protected. R. Out[]=. ote again that Mathematica struggles to perform the sum numerically, but still suggests that the remainder shrinks with increasing. ow consider the 4 values used above In[]:= Plot@8R@,.3D, R@,.D, R@,.7D, R@,.9D<, 8,, <, PlotRange 8, <, AesLabel 8, R<, AesOrigin -> 8, <D R. Out[]=. 3 4 As epected all the remainders head to zero for, although even on the current scale the remainder has only shrunk to about. at =. Clearly this is not very satisfactory and is why, even with computers available, we also need the analytic tests for convergence discussed in Lecture.

6 6 Lec_7_8.nb As epected all the remainders head to zero for, although even on the current scale the remainder has only shrunk to about. at =. Clearly this is not very satisfactory and is why, even with computers available, we also need the analytic tests for convergence discussed in Lecture. What about the case = In[3]:= Plot@R@,.D, 8,, <, PlotRange 8, <, AesLabel 8, R<, AesOrigin -> 8, <D General::stop : Further output of Sum::div will be suppressed during this calculation. R. Out[3]=. So in this case Mathematica knows that something is wrong. Another way to look at is in terms of the limit, (where we epect a zero answer for convergence) In[4]:= Out[4]= Limit@R@,.D, D. So Mathematica does not handle this well. On the other hand once we have "summed" the general sum In[]:= R@, D + Out[]= - We can use this form to take the large limit for fied easily (note the substitution command /.>value) + In[6]:= - + Out[6]= Take the limit + In[7]:= LimitB- + Out[7]=...9, F So the series converges. At the boundary = we have

7 Lec_7_8.nb + In[8]:= LimitB- +.., F Power::infy : Infinite epression encountered.. Out[8]= CompleInfinity So this epression clearly diverges, and for > + In[9]:= LimitB- + Out[9]= -.., F We can also see this behavior in the plot of the summed series S[], i.e., the divergence as approaches. In[3]:= Plot@S@D, 8, -, <, AesLabel 8, S<D General::stop : Further output of Sum::div will be suppressed during this calculation. S 4 3 Out[3]= - - So if we carefully evaluate the infinite series representing the remainder and takes its limit,, Mathematica will accurately tell us about the convergence properties of the series. On the other hand, we often have difficulty proceeding just numerically. The analytic analysis we have discussed in the Lecture is essentially for proceeding either by hand or via Mathematica. To make use of series, we need to understand when they make sense and when they do not via analytic methods. 7