Use a series to evaluate to four decimal places.
SOLUTION: Although cannot be expressed in terms of elementary functions, we can write
a series for eu, replace u by (−x^2 ), and integrate term by term. Thus,Since this is a convergent alternating series (with terms decreasing in magnitude and approaching
0), which will not affect the fourth decimal place. Then, correct to four decimal places,†C7. Power Series over Complex Numbers.
A complex number is one of the form a + bi, where a and b are real and i^2 = −1. If we allow complex
numbers as replacements for x in power series, we obtain some interesting results.
Consider, for instance, the series
When x = yi, then (1) becomes
Then
since the series within the parentheses of equation (2) converge respectively to cos y and sin y.
Equation (3) is called Euler’s formula. It follows from (3) that
ei π = − 1,
and thus that
ei π + 1 = 0,