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Notebook[{
Cell[TextData[{
  StyleBox["MATH 257 Calculus III\t\t\t\t\t\t\t\tWeek of April 7, 2003",
    FontSize->14],
  "\n",
  StyleBox["Lab 2: Serious Series", "Title"],
  "\n",
  StyleBox["Name 1\nName 2\nSection:", "Section"]
}], "Text"],

Cell["\<\
I.  Laboratory Objectives
\tto get down with series, geometric series, 
\tand the integral test\
\>", "Text",
  FontSize->14],

Cell[TextData[{
  "II.  Formatting and Syntax Information:\n\tUse the Basic Input Palette to \
find ",
  StyleBox[" \n\t", "Section"],
  Cell[BoxData[
      \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] a\_n\)],
    FontSize->18],
  StyleBox["    and   ",
    FontSize->18],
  Cell[BoxData[
      \(TraditionalForm\`\[Integral]\_a\%b f[x] \[DifferentialD]x\)],
    FontSize->18]
}], "Text",
  FontSize->14],

Cell[TextData[StyleBox["Infinite Series",
  FontSize->20]], "Title",
  Evaluatable->False,
  AspectRatioFixed->False,
  FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
  "An infinite series is an infinite sum.  \nIn the",
  StyleBox[" ",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox["infinite sequence",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  ", {",
  Cell[BoxData[
      \(TraditionalForm\`a\_n\)]],
  "} represents an infinite list of values:\n\t{",
  Cell[BoxData[
      \(TraditionalForm\`\(a\_1, \ a\_2, \ a\_3, \ \(.\ .\ .\)\ }\)\)]],
  "\nIn an ",
  StyleBox["infinite series",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] a\_n\)]],
  " represents an infinite sum:\n\t",
  Cell[BoxData[
      \(TraditionalForm\`a\_1 + a\_2 + a\_3 + \(.\ .\ .\)\)]],
  "\n\t\nSummations can be finite or infinite.  Let's start with a finite \
sum.  The sum below has ten terms being added.  Execute the statement below:\t\
"
}], "Text"],

Cell[BoxData[
    \(\[Sum]\+\(n = 1\)\%10\( 5 - n\)\/\(3  n\^5\)\)], "Input",
  FontSize->14],

Cell[TextData[{
  "To find the decimal approximation of this fraction to 20 significant \
digits, use the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " command below.  The syntax is ",
  StyleBox["N",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" for numeric value",
    FontWeight->"Bold"],
  ", the ",
  StyleBox["%",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" sign refers to the previous computation",
    FontWeight->"Bold"],
  " and ",
  StyleBox["20",
    FontWeight->"Bold",
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[" is the number of significant digits",
    FontWeight->"Bold"],
  " you desire when computing the numeric value."
}], "Text"],

Cell[BoxData[
    \(N[%, 20]\)], "Input",
  FontSize->14],

Cell[TextData[{
  "What you have now calculated is the 10th partial sum or ",
  Cell[BoxData[
      \(TraditionalForm\`S\_10\)]],
  "of the infinite series ",
  Cell[BoxData[
      \(\[Sum]\+\(n = 1\)\%\[Infinity]\( 5 - n\)\/\(3  n\^5\)\)]],
  ".  Suppose you wanted the sum of the first 150 terms of this infinite \
series:"
}], "Text"],

Cell[BoxData[
    \(\[Sum]\+\(n = 1\)\%150\( 5 - n\)\/\(3  n\^5\)\)], "Input",
  FontSize->14],

Cell[TextData[{
  "This huge fraction is hard to read and even harder to estimate, so let's \
once again ask ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " for a decimal approximation:"
}], "Text"],

Cell[BoxData[
    \(N[%, 20]\)], "Input",
  FontSize->14],

Cell[TextData[{
  "It seems that our series is converging. ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can check this for us."
}], "Text"],

Cell[BoxData[
    \(\[Sum]\+\(n = 1\)\%\[Infinity]\( 5 - n\)\/\(3  n\^5\)\)], "Input",
  FontSize->14],

Cell["\<\
This is not much help unless you are familiar with the Zeta \
function.  Let's ask for a decimal approximation:\
\>", "Text"],

Cell[BoxData[
    \(N[%, 20]\)], "Input",
  FontSize->14],

Cell[TextData[{
  "\nAnother way to see what is going on would be to make a ",
  StyleBox["ListPlot", "Input"],
  " of the ",
  StyleBox["sequence of partial sums",
    FontWeight->"Bold"],
  " ",
  StyleBox["S[n]", "Input"],
  ".  The following will do this:"
}], "Text"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(S[
        n_] := \[Sum]\+\(k = 1\)\%n\( 5 - k\)\/\(3  k\^5\)\)\)\)], "Input"],

Cell[TextData[{
  "\nSee how we had to use k instead of n there?  That's because we wanted \
the sum to go ",
  StyleBox["from 1 to n",
    FontWeight->"Bold"],
  ", so we needed a second variable to keep track of this.  Now we can ",
  StyleBox["ListPlot", "Input"],
  " it using the ",
  StyleBox["Table", "Input"],
  " command.  Let's plot it for n=1 to 100:"
}], "Text"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(ListPlot[
      Table[{n, S[n]}, {n, 1, 100}]]\)\)\)], "Input"],

Cell[TextData[StyleBox["Geometric Series",
  FontSize->20]], "Title",
  Evaluatable->False,
  AspectRatioFixed->False,
  FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
  "Geometric Series are a specific type of infinite series.  Geometric Series \
have the form:\n\n\t",
  StyleBox["a +  ar  +  ",
    FontWeight->"Bold"],
  Cell[BoxData[
      \(TraditionalForm\`ar\^2\  + \ ar\^3\  + \ 
          ar\^\(\(4\)\(\ \)\) + \ \(\(.\)\(\ \)\(.\)\(\ \)\(.\)\)\ \  = \ \
\[Sum]\+\(n = 1\)\%\[Infinity] ar\^\(n - 1\)\)],
    FontWeight->"Bold"],
  "\n\n\t",
  StyleBox["a",
    FontColor->RGBColor[1, 0, 0]],
  " is the first term of the series and can be any real number\n\t",
  StyleBox["r",
    FontColor->RGBColor[1, 0, 0]],
  " is the ratio of adjacent terms.   It can also be any real number. "
}], "Text"],

Cell[TextData[{
  StyleBox["\n",
    FontSize->20],
  StyleBox["When | r | < 1, the Geometric Series converges, but\nwhen | r | \
\[GreaterEqual] 1, the Geometric Series diverges.\nThe sum of a convergent \
Geometric Series is ",
    FontSize->18],
  Cell[BoxData[
      \(TraditionalForm\`a\/\(1 - r\)\)],
    FontSize->24]
}], "Title",
  Evaluatable->False,
  AspectRatioFixed->False,
  FontColor->RGBColor[0, 0, 1]],

Cell[BoxData[{
    RowBox[{
      RowBox[{
        StyleBox[\(Let'\),
          "Section",
          FontSize->18], 
        StyleBox["s",
          "Section",
          FontSize->18], 
        StyleBox[" ",
          "Section",
          FontSize->18], 
        StyleBox["look",
          "Section",
          FontSize->18], 
        StyleBox[" ",
          "Section",
          FontSize->18], 
        StyleBox["at",
          "Section",
          FontSize->18], 
        StyleBox[" ",
          "Section",
          FontSize->18], 
        StyleBox["a",
          "Section",
          FontSize->18], 
        StyleBox[" ",
          "Section",
          FontSize->18], 
        StyleBox["few",
          "Section",
          FontSize->18], 
        StyleBox[" ",
          "Section",
          FontSize->18], 
        RowBox[{
          StyleBox["examples",
            "Section",
            FontSize->18], 
          StyleBox[":",
            "Section",
            FontSize->18], "\[IndentingNewLine]", 
          StyleBox[
            RowBox[{"The", " ", "series", "         ", 
              RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), 
                StyleBox[\(4 \((3\/5)\)\^\(n - 1\)\),
                  "Subsection",
                  FontSize->14]}]}],
            FontFamily->"Times"]}]}], 
      StyleBox[" ",
        "Subsection",
        FontFamily->"Times",
        FontSize->14]}], "\n", 
    StyleBox[\(has\ a\  = \ \(4\ and\ r\  = \ 
          3/5. \ Since\  | r\  | \ \(\(<\)\(\ \)\(1\)\)\), \ 
      this\ series\ will\ converge . \nLet' 
        s\ look\ at\ the\ nth\ partial\ sums . \ \ First, \ 
      define\ the\ sequence\ of\ partial\ sums, \ 
      and\ then\ evaluate\ \(\(it\)\(:\)\)\),
      FontFamily->"Times"]}], "Text",
  FontSize->14],

Cell[BoxData[
    RowBox[{\(S[n_]\), ":=", " ", 
      RowBox[{\(\[Sum]\+\(k = 1\)\%n\), 
        StyleBox[\(4 \((3\/5)\)\^\(k - 1\)\),
          "Subsection",
          FontSize->14]}]}]], "Input",
  FontSize->12],

Cell[BoxData[
    \(S[1]\)], "Input",
  FontSize->14],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(S[2]\)\)\)], "Input"],

Cell[BoxData[
    \(N[%]\)], "Input"],

Cell[BoxData[
    \(S[5]\)], "Input"],

Cell[BoxData[
    \(N[%]\)], "Input"],

Cell[BoxData[
    \(S[10]\)], "Input"],

Cell[BoxData[
    \(N[%]\)], "Input"],

Cell["\<\

Try getting this next one as a decimal to 20 significant digits:\
\>", "Text"],

Cell[BoxData[
    \(S[50]\)], "Input"],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[BoxData[
    \(S[100]\)], "Input"],

Cell[BoxData[
    \(N[%, 30]\)], "Input"],

Cell[BoxData[
    \(S[n]\)], "Input"],

Cell[TextData[StyleBox["To find the limit of the nth partial sum:",
  FontWeight->"Bold"]], "Text",
  FontFamily->"Helvetica",
  FontSize->14],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(Limit[S[n], \ 
      n \[Rule] \ \[Infinity]]\)\)\)], "Input"],

Cell[TextData[{
  StyleBox["Of course, we could have asked ", "Section"],
  StyleBox["Mathematica", "Section",
    FontSlant->"Italic"],
  StyleBox[" to compute this infinite sum directly:", "Section"]
}], "Text",
  FontSize->14,
  FontWeight->"Bold"],

Cell[BoxData[
    RowBox[{" ", 
      RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), 
        StyleBox[\(4 \((3\/5)\)\^\(n - 1\)\),
          "Subsection",
          FontSize->14]}]}]], "Input"],

Cell[TextData[{
  StyleBox["The Integral Test on ",
    FontSize->20],
  StyleBox["Mathematica",
    FontSize->20,
    FontSlant->"Italic"],
  StyleBox[" ",
    FontSize->20]
}], "Title",
  Evaluatable->False,
  AspectRatioFixed->False,
  FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
  "Many infinite series are not geometric, and so a large portion of this \
course is spent developing tests for infinite series which tell us if the \
series converges or diverges. When geometric series converge, we can always \
find the sum of the series: S = ",
  Cell[BoxData[
      \(TraditionalForm\`a\/\(1 - r\)\)]],
  ".  For most series, however, we will only be able to tell if the series \
converges or diverges . .  finding the sum of the series is often just too \
difficult by hand.\n\nIntegral Test:\nIf f(x) is a continuous, positive and \
decreasing function on [1, \[Infinity]) and \n",
  Cell[BoxData[
      \(TraditionalForm\`a\_n\)]],
  "= f(n) for each term in the positive termed series ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] a\_n\)]],
  ",\nthen ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] a\_n\)]],
  " converges if and only if ",
  Cell[BoxData[
      \(TraditionalForm
      \`\(\[Integral]\_1\%\[Infinity]\( f(x)\) \[DifferentialD]x\ \)\)]],
  "converges.\n\nLook at the example ",
  Cell[BoxData[
      \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\^2\)]],
  ".  \nFirst check that the corresponding function f(x) = ",
  Cell[BoxData[
      \(TraditionalForm\`1\/x\^2\)]],
  "is continuous, positive and decreasing on [1,\[Infinity]) by graphing \
f(x)."
}], "Text"],

Cell[BoxData[
    \(Plot[1\/x\^2, \ {x, 0, 20}]\)], "Input"],

Cell["\<\
Then check if the integral from 1 to \[Infinity] converges or \
not:\
\>", "Text"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\[Integral]\_1\%\[Infinity]\( 
            1\/x\^2\) \[DifferentialD]x\)\)\)], "Input"],

Cell[TextData[{
  "Since the improper integral ",
  Cell[BoxData[
      \(TraditionalForm\`\[Integral]\_1\%\[Infinity]\( 
              1\/x\^2\) \[DifferentialD]x\)]],
  " is convergent, the infinite series \n",
  Cell[BoxData[
      \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\^2\)]],
  "is also convergent.\n\n",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " can actually compute the sum of the series:"
}], "Text"],

Cell[BoxData[
    \(\[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\^2\)], "Input"],

Cell[TextData[{
  "Problems \nThis is the part of the lab you will hand in for the lab.  ",
  StyleBox["Delete all the above stuff (except the title, your name, and \
section) before you submit your lab in Blackboard.",
    FontColor->RGBColor[0, 0, 1]],
  " "
}], "Subtitle",
  FontSize->24],

Cell[TextData[{
  StyleBox["1.  Given the series  ",
    FontColor->RGBColor[1, 0, 0]],
  Cell[BoxData[
      \(\[Sum]\+\(n = 1\)\%\[Infinity] 6 \((5\^\(-n\))\)\)],
    FontColor->RGBColor[1, 0, 0]],
  StyleBox[":",
    FontColor->RGBColor[1, 0, 0]]
}], "Text",
  FontSize->18],

Cell[TextData[{
  "\ta.)  Find the 5th partial sum, ",
  Cell[BoxData[
      \(TraditionalForm\`S\_5\)]],
  " and its decimal approximation."
}], "Text",
  FontSize->14,
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "\tb.)  Find the 10th partial sum, ",
  Cell[BoxData[
      \(TraditionalForm\`S\_10\)]],
  " and its decimal approximation to 10 significant digits."
}], "Text",
  FontSize->14,
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "\tc. )  Find the 20th partial sum, ",
  Cell[BoxData[
      \(TraditionalForm\`S\_20\)]],
  " and its decimal approximation to 20 significant digits"
}], "Text",
  FontSize->14,
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell["\td.)  What is the sum of the infinite series ?", "Text",
  FontSize->14,
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "\te.)  Identify ",
  StyleBox["a",
    FontSlant->"Italic"],
  " and ",
  StyleBox["r",
    FontSlant->"Italic"],
  " for this series and compare the answer ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " just gave you with the computation of ",
  Cell[BoxData[
      \(TraditionalForm\`a\/\(1\  - \ r\)\)]],
  "."
}], "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "2.  Given the series ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          StyleBox[
            SuperscriptBox[
              RowBox[{
                FormBox[\(\[Sum]\+\(n = 1\)\%\[Infinity]\ \((4\/3\)\),
                  "TraditionalForm"], ")"}], "n"],
            FontWeight->"Bold"], "  "}], TraditionalForm]]],
  " "
}], "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
  "\ta. )  Identify the first term ",
  StyleBox["a",
    FontSlant->"Italic"],
  " and the ratio ",
  StyleBox["r",
    FontSlant->"Italic"],
  "."
}], "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell["\<\
\tb.)  Do you think the infinite series converges or diverges?  \
Why?\
\>", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "\tc.)  Find the 20th partial sum ",
  Cell[BoxData[
      \(TraditionalForm\`S\_20\)]],
  " and its decimal approximation to 15 significant digits",
  Cell[BoxData[
      \(TraditionalForm\` . \)]]
}], "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "\td.)  Find the 100th partial sum",
  Cell[BoxData[
      \(TraditionalForm\`S\_100\)]],
  " and its decimal approximation to 15 significant digits."
}], "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "3. Given the following series: ",
  StyleBox[" ",
    FontSize->18],
  StyleBox[" 1 - ",
    FontSize->24],
  Cell[BoxData[
      \(TraditionalForm\`2\/3\)],
    FontSize->24],
  StyleBox[" + ",
    FontSize->24],
  Cell[BoxData[
      \(TraditionalForm\`4\/\(\(9\)\(\ \)\)\)],
    FontSize->24],
  StyleBox["- ",
    FontSize->24],
  Cell[BoxData[
      \(TraditionalForm\`8\/27\)],
    FontSize->24],
  StyleBox[" ",
    FontSize->24],
  StyleBox["+ . . .",
    FontSize->18]
}], "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell["\ta. Write this infinite series using summation notation.", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "\t",
  StyleBox["b.  Find the 10th partial sum, ",
    FontWeight->"Plain"],
  Cell[BoxData[
      \(TraditionalForm\`S\_10\)],
    FontWeight->"Plain"],
  StyleBox[" and its decimal approximation.",
    FontWeight->"Plain"]
}], "Text",
  FontFamily->"Helvetica",
  FontSize->14,
  FontWeight->"Bold",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  StyleBox["\tc.  Find the 20th partial sum ",
    FontWeight->"Plain"],
  Cell[BoxData[
      \(TraditionalForm\`S\_\(\(20\)\(\ \)\)\)],
    FontWeight->"Plain"],
  StyleBox["and its decimal approximation to 20 significant digits.  ",
    FontWeight->"Plain"]
}], "Text",
  FontFamily->"Helvetica",
  FontSize->14,
  FontWeight->"Bold",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[StyleBox["\td.  At this point, what would you guess as the sum \
of the infinite series?",
  FontWeight->"Plain"]], "Text",
  FontFamily->"Helvetica",
  FontSize->14,
  FontWeight->"Bold",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  StyleBox["\te.  Make a ListPlot of the partial sums ",
    FontWeight->"Plain"],
  Cell[BoxData[
      \(TraditionalForm\`S\_n\)],
    FontWeight->"Plain"],
  StyleBox[" from n=1 to 50.",
    FontWeight->"Plain"]
}], "Text",
  FontFamily->"Helvetica",
  FontSize->14,
  FontWeight->"Bold",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "\t",
  StyleBox["f.  Since this is a geometric series, identify ",
    FontWeight->"Plain"],
  StyleBox["a",
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox[" and ",
    FontWeight->"Plain"],
  StyleBox["r",
    FontWeight->"Plain",
    FontSlant->"Italic"],
  StyleBox[".  Use the formula for the sum of a convergent geometric series \
to find the sum of this series.",
    FontWeight->"Plain"]
}], "Text",
  FontFamily->"Helvetica",
  FontSize->14,
  FontWeight->"Bold",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  StyleBox["4.  Given the series ",
    FontSize->18],
  Cell[BoxData[
      FormBox[
        RowBox[{
          StyleBox[
            UnderoverscriptBox["\[Sum]", 
              RowBox[{"n", 
                StyleBox["=",
                  FontSize->14], 
                StyleBox["2",
                  FontSize->14]}], "\[Infinity]"],
            FontSize->24], 
          SuperscriptBox[
            StyleBox[
              RowBox[{
                StyleBox["n",
                  FontSize->24], "e"}]], \(-n\)]}], TraditionalForm]],
    FontSize->18]
}], "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[TextData[{
  "\ta.  Use the Plot command to show that f(x) = ",
  Cell[BoxData[
      FormBox[
        SuperscriptBox[
          StyleBox[
            RowBox[{
              StyleBox["x",
                FontSize->18], "e"}]], \(-x\)], TraditionalForm]]],
  "is a continuous, decreasing and positive function on the interval [2, \
100]",
  StyleBox[".",
    FontWeight->"Bold"],
  "  "
}], "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell["\<\
\tb.  Now use Integral Test to show whether or not the series \
converges.  \
\>", "Text",
  FontColor->RGBColor[1, 0, 0]],

Cell[BoxData[
    \(\[IndentingNewLine]\)], "Input"],

Cell[TextData[{
  "\tc.  Does the series  ",
  Cell[BoxData[
      FormBox[
        RowBox[{
          StyleBox[
            UnderoverscriptBox["\[Sum]", 
              RowBox[{"n", 
                StyleBox["=",
                  FontSize->14], 
                StyleBox["2",
                  FontSize->14]}], "\[Infinity]"],
            FontSize->24], 
          SuperscriptBox[
            StyleBox[
              RowBox[{
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