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Taylor Series Calculus

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The starting point for this example is the Taylor series for sine:
sin (x) = x - (1/3!)x^3 + (1/5!)x^5 ?(1/7!)x^7 + ...... + (-1)^n * 1/(2n + 1)!*x^(2n+1) + .......
a) Let f(x) = { sin(x) / x if x doesn't equal 0
{1 if x doesn't equal 0

Show that f is infinitely differentiable on R (including x = 0) and determine its Taylor series in powers of x.
b) Define the function Si by the rule,
Si (x) = the integral from 0 to x
sin(t) / t dt , x E R
Determine the Taylor series for Si in powers of x, and the open interval of convergence of the resulting series.

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Solution Preview

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The starting point for this example is the Taylor series for sine:
sin (x) = x - (1/3!)x^3 + (1/5!)x^5 -(1/7!)x^7 + ...... + (-1)^n * 1/(2n + 1)!*x^(2n+1) + .......
a) Let f(x) = { sin(x) / x if x doesn't equal 0
{1 if x doesn't equal 0

Show that f is infinitely differentiable on R (including x = 0) and determine its Taylor series in powers of x.
b) Define the function Si by the rule,
Si (x) = the integral from 0 to x
sin(t) / t ...

Solution Summary

The solution provides an example of the Taylor series caluclus.

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