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Blast into Math! Analatic nummer theora: ants, ghosts and giants

is also monotonically increasing and bounded sequence by the same argument. Therefore,
−S converges. Let’s call the limit L . Then, since −S converges to L , S converges
−
−
−
−
−
N N N
to −L .
−
Exercise: Prove this using the definition of convergence of a sequence.
+
By the LAMP, S + S converges to L − L .
+
−
−
N
N
Now, notice that the series which define the functions sin(x) and cos(x) are similar to the
series which defines e . So, you can prove that for each x ∈ R the series which define sin(x)
x
and cos(x) converge by comparing them to the series which is used to define the function
e . Use the definition of
x
∞ n
x
x
e =
n!
n=1
to define

iπ
e .

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