Solutions

In[83]:=

(*1*)Series[Sin[x]/x, {x, 0, 5}]

Out[83]=

1 - x^2/6 + x^4/120 + O[x]^6

The function Normal cuts of the O[x^n] term from the series so that it can be plotted.

In[84]:=

(*2*)s1 = Series[Sin[x], {x, 0, 2}]//Normal ; s2 = Series[Sin[x], {x, 0, 5}] ... [{Sin[x], s1, s2, s3}, {x, -2Pi, 2Pi}, PlotStyle {Hue[0], Hue[1/4], Hue[1/2], Hue[3/4]}]

[Graphics:../HTMLFiles/ex04_solutions_129.gif]

Out[87]=

⁃Graphics⁃

When the expanded function is undefined, you get the general form of the Taylor's series in terms of the derivatives of the function

In[88]:=

(*3*)Clear[f] ; Series[f[x], {x, y, 5}]

Out[89]=

f[y] + f^′[y] (x - y) + 1/2 f^′′[y] (x - y)^2 + 1/6 f^(3)[y] (x - y)^3 + 1/24 f^(4)[y] (x - y)^4 + 1/120 f^(5)[y] (x - y)^5 + O[x - y]^6

In[90]:=

(*4*)Series[Tan[x], {x, Pi/2, 5}] Series[Sin[x], {x, Pi/2, 7}]/Series[Cos[x], {x, Pi/2, 7}]

Out[90]=

-1/(x - π/2) + 1/3 (x - π/2) + 1/45 (x - π/2)^3 + 2/945 (x - π/2)^5 + O[x - π/2]^6

Out[91]=

-1/(x - π/2) + 1/3 (x - π/2) + 1/45 (x - π/2)^3 + 2/945 (x - π/2)^5 + O[x - π/2]^6

The Bessel function has an essential singularity in infinity (and in an infinite number of other points). For example, the function
    1/(Underscript[∑, i]x^i)
has an essential singularity at x=0. There are terms with an infinite negative exponent.

In[92]:=

(*5*)Series[BesselJ[0, x]^2, {x, Infinity, 5}]//NormalPlot[{%, BesselJ[0, x]}, {x, 1000, 1020}, PlotStyle {Hue[0], Hue[2/3]}]

RowBox[{Series :: esss, : , RowBox[{Essential singularity encountered in , RowBox[{Cos, [, I ... ta[x, Infinity, {-1, Rational[1, 4] Pi}, -1, 8, 1], Editable -> False], ]}], . , More…}]}]

RowBox[{Series :: esss, : , RowBox[{Essential singularity encountered in , RowBox[{Cos, [, I ... eriesData[x, 0, {-1, Rational[1, 4] Pi}, -1, 8, 1], Editable -> False], ]}], . , More…}]}]

RowBox[{Series :: esss, : , RowBox[{Essential singularity encountered in , RowBox[{Cos, [, I ... ta[x, Infinity, {-1, Rational[1, 4] Pi}, -1, 9, 1], Editable -> False], ]}], . , More…}]}]

General :: stop : Further output of Series :: esss will be suppressed during this calculation. More…

Out[92]=

((2/π^(1/2) 1/x^(1/2) - (9 (1/x)^(5/2))/(64 (2 π)^(1/2)) + (3675 (1/x)^(9/2))/(163 ... ))/(512 (2 π)^(1/2)) - (59535 (1/x)^(11/2))/(131072 (2 π)^(1/2))) Sin[π/4 - x])^2

[Graphics:../HTMLFiles/ex04_solutions_144.gif]

Out[93]=

⁃Graphics⁃

In[94]:=

(*6*)Clear[m, v, c] ; m[v_] = m_0/Sqrt[1 - (v/c)^2] ; Series[m[v] c^2, {v, 0, 3}]

Out[96]=

c^2 m_0 + (m_0 v^2)/2 + O[v]^4

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